Total 56, integers are remained if all multiples of 4 and all multiples of 5 are removed from the set of integers from 1 through 100.
We have given that,
from the set of first 100 integers numbers, all the multiples of 5 and 4 are removed.
According to the question,
The multiples of 4 upto 100,
S = { 4, 8, 12......, 96}
it is an Arithmetic progression sequence with
a = 4 , l = 96 , d = 4
using an = a + (n-1)d where
n --> number of terms in sequence
d ---> common difference between two consecutive terms
a ---> first term
=> 96 = 4 + (n-1)4
=> (n-1) = 92/4
=> n = 23 + 1 = 24
Total multiples of 4 upto 100 is 24 .
Now, multiples of 5 upto 100
Q = { 5, 10, 15, ....,100}
Similarly as for above case
a = 5 , d = 5 , l = 100
=> 100 = 5 + (n-1)5
=> (n-1)5 = 95
=> n-1 = 19
=> n = 20
Total multiples of 5 upto 100 is 20.
Remained integers upto 100 = 100 - 20 -24
= 56
Hence, the required number of integers are 56.
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a rectangular backyard pool measures 12m by 4m the pathway around the outside of it is 80cm wide. what is the outside perimeter of the pool area
The Perimeter of the Pathway around the pool is 38.4m
What is Perimeter?
A perimeter is a closed route that covers, surrounds, or outlines a two-dimensional form or length. The circumference of a circle or an ellipse is its perimeter. There are various practical applications for calculating the perimeter.
Solution:
Length of the Pathway = 12m + 80cm + 80cm = 12m + 1.6m = 13.6m
Breadth of the Pathway = 4m + 80cm + 80 cm = 4m + 1.6m = 5.6m
Perimeter of Rectangle = 2(l+b)
Perimeter = 2(13.6 + 5.6)
Perimeter = 2(19.2)
Perimeter = 38.4m
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tell whether each point is on the graph of f(x)=|x|.If it is, give the coordinates of another point with the same y value. (11,11)
The point (11,11) will lie on the graph of the function f(x) = IxI
What is coordinate?
The x-coordinate informs us of a point's separation from the y-axis, which is the vertical axis. The abscissa is another term for the x-coordinate. The y-coordinate shows how far a point is from the horizontal axis, or x-axis. The term "ordinate" also applies to the y-coordinate.
the function: f(x) = IxI
The function will have every positive and negative integer contained within itself.
So coordinates of another point with the same y value (11,11) will lies on the graph so as (-11,11).
Hence point (11,11) will lie on the graph of the function f(x) = IxI
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What is the value of x?
The value of x in the diagram is 9
What is an equilateral triangle?equilateral triangle is a triangle that has all its sides equal in length. Since the three sides are equal therefore the three angles, opposite to the equal sides, are equal in measure. Therefore, it is also called an equiangular triangle, where each angle measure 60 degrees. Just like other types of triangles, an equilateral triangle also has its area, perimeter and height formula
From the diagram all the sides are equal which means it is an equilateral triangle in which each angle is equal to 60°
To find x,
7x -3 = 60
7x = 60 + 3
7x = 63
x = 63/7
x = 9
In conclusion the value of x is 9
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when will the height be 37 feet?
As the tree grows 1& 3/4 feet per year, it will take 9 years to reach a height of 37 feet.
What is division?Division is a basic procedure in which an integer is divided. It's best to imagine of it as a number of things being distributed among a particular number of persons, such as in the previous example. In mathematics, division is the process of dividing a number into equal parts and determining how many equal parts can be made. For instance, dividing 15 by 3 implies dividing 15 into three equal groups of five. Dividend Divisor = Quotient + Remainder is the formula for dividing two numbers. The dividend is the number that is being divided in this case. The divisor is the number, which divides the number (dividend) into equal parts.
Here,
37 - 21 and 1/4 = 36 and 4/4 - 21 and 1/4 = 15 and 3/4 = 63/4
63/4 divided by 1 and 3/4 = 63/4 divided by 7/4 = 63/4 * 4/7 = 9 years
It will take 9 years to reach at the height of 37 feet as the tree grow 1& 3/4 feet per year.
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Complete question:
A tree grow 1& 3/4 feet per year. How long will it take the tree to grow from a height of 21 & 1/4 feet to a height of 37 feet?
which linear equation represents a line with a slope of 9494 and a y-y-intercept of 0?
The required equation of the line is 9/4x with a slope of 9/4 and a y-intercept of 0.
What is the slope of the line?The value of the steepness or the direction of a line in a coordinate plane is known as a line's slope, often referred to as a line's gradient. With a line's equation or the coordinates of a point along the straight line, slope can be computed using a variety of ways.
The slope of a line indicates the steepness and direction of the line. Finding the difference between the coordinates of the locations will allow you to quickly compute the slope of a straight line connecting two points, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The letter "m" is typically used to signify slope.
The slope of the line is a tangent angle made by line with horizontal.
i.e. m = tan x, where x in degrees.
Let the given values be m = 9/4 and c = 0
Standard equation of the line, y = mx + c
substitute the values in the above equation, we get
y = 9/4x + 0
y = 9/4x
Therefore, the required equation of the line exists 9/4x.
The complete question is:
Which linear equation represents a line with a slope of 9/4 and a y-intercept of 0?
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below are drawings of a wide and a narrow cylinder. the cylinders have equally spaced marks on them. water is poured into the wide cylinder up to the 4th mark (see a). this water rises to the 6th mark when poured into the narrow cylinder (see b). both cylinders are emptied, and water is poured into the narrow cylinder up to the 11th mark. how high would this water rise if it were poured into the empty wide cylinder?
The answer is to the 7 1/3 th mark.
Consider the provided question,
Since, in wide cylinder, water rises to 4th mark while in narrow cylinder same amount of water rises to 6th mark.
So, 6th mark on narrow cylinder = 4th mark on the wide cylinder which implies that 3rd on narrow cylinder = 2th mark on the wide cylinder
From the above,
Since, 3rd on narrow cylinder = 2th mark on the wide cylinder that means 1 mark on narrow cylinder is equal to 2/3 the of the 1st mark of the wider cylinder.
Hence the 11th mark of the narrow cylinder is equal to the
11 * 2/3 = 22/3 = 7 1/3 mark of the wider cylinder.
Hence, the correct answer is option to the 7 1/3th mark.
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PLEASE HELP ME WITH THESE 2 QUESTIONS
Answer:
Question 1: Calculate the coordinates of 2 points. You know that the value of the Y coordinate in a point is 4 times the value of the x in the same point. Once you figure out the points, put them on the graph and draw a straight line through them with a ruler.
Question 2: What number do you multiply 5 by to get 12? You multiply that same number by the 375 to get the value in the corresponding row. That's called proportion.
Question 3: You can find the proportion between the two values (distance and energy). Its how many times bigger (or smaller) the values are in relation to each other. You can use this proportion for the scale of the graph so that the location of points are easier understood in relation to each other and discernable all by your eye.
decide whether enough information is given to prove that $\triangle abc\cong\triangle dbe$ using the sss congruence theorem. explain. two triangles, triangle a c b and triangle d e b that share a common vertex b. point b is on the segment a d. in triangle a c b, side a c is marked with single tick, side b c is marked with double ticks and side a b is marked with three ticks. in triangle d e b, side d e is marked with single tick, side b e is marked with double ticks and side b d is marked with three ticks.put responses in the correct input to answer the question. select a response, navigate to the desired input and insert the response. responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. responses can also be moved by dragging with a mouse.you are given that $\overline{ab}\cong$ response area, $\overline{bc}\cong$ response area, and $\overline{ac}\cong$ response area. so, the triangles response area be proven congruent using the sss congruence theorem.
No, enough information is not given to prove that triangles are congruent using the SSS congruence theorem.
The SSS congruence theorem expresses that assuming in two triangles, three sides of one are consistent to three sides of the other, then, at that point, the two triangles are said to be congruent.
Now, we are given that triangle ACB and triangle DEB that share a common vertex b. So, both triangles share a common side. So, one side is same for both of triangles.
From the diagram, we can see that both triangles opposite side are equal. So, second side is equal for both of triangles.
Now, from the diagram, we can see that both triangles vertically opposite angles are equal. So, we have two sides equal and one angle is equal.
So, given triangle is proved to congruent by SAS congruency not by SSS congruency.
Hence, enough information is not given.
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A given measures 30 degrees, and the measure of its vertical angle, is expressed as (5x+5) degrees. Solve for x
After solving the equation, the value obtained for x will be equal to 5.
What is an expression?Mathematical actions are called expressions if they have at least two terms that are related by an operator and include either numbers, variables, or both. Adding, subtraction, multiplying, and division are all reflection coefficient operations.
A mathematical operation such as reduction, addition, multiplication, or division is used to integrate terms into an expression.
As per the given expression in the question,
5x + 5 = 30
5x = 30 - 5
5x = 25
x = 25/5
x = 5
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question which system of equations models the following problem if x represents the number of angelfish yves bought and y represents the number of parrotfish he bought? yves bought 420 tropical fish for a museum display. he bought 6 times as many parrotfish as angelfish. how many of each type of fish did he buy?
Thus, 420 fish were ultimately caught, with 6x standing in for "6 times as much."
Calculation:x + y = 420; y = 6x
Thus, 420 fish were ultimately caught, with 6x standing in for "6 times as much."
How well-versed are you in fish size?The smallest and largest fish range in size from 1 cm to 18 m. 18 m then equals 18 x 100 or 1800 cm. ∴ The length of large fish exceeds that of little fish by 1800 times.
How many parrotfish purchased by Carlos?The number of parrotfish Carlos purchased is represented by the variable y, whereas the number of angelfish Carlos purchased is represented by the variable x.
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which statement describes whether marjan’s graphs are correct
Yes, g(x) is a 3 unit translation down after a reflection of f(x) over the x-axis.
What is meant by graphs?The set of ordered pairs with the f(x) = y is the graph of a function f in mathematics. These pairs, which are Cartesian coordinates of points in two-dimensional space and thus constitute a subset of this plane in the typical case where x and f(x) are real numbers, are examples of real numbers.
Pie charts, line graphs, bar graphs and histograms, and Cartesian graphs are likely the four most popular types. They are typically utilized for, and are most effective for, very different purposes. You'd employ: Bar graphs are used to display numbers that are unrelated to one another.
Use line/area charts if your data is continuous, or bar charts or histograms if your data is nominal. Use a scatter plot, bubble chart, or line chart to illustrate how the values in your dataset relate to one another.
y = −f(x) Reflects it about x-axis
y = f(x) + C C < 0 moves it down
The complete question is:
Consider the function f(x) = x3 + 0.5x – 1. Marjan needs to graph f(x) and g(x) = –f(x) – 3. His graphs are shown below. Which statement describes whether Marjan’s graphs are correct? No, the reflection should be over the x-axis, not the y-axis. No, the reflection should be over the y-axis, not the x-axis. Yes, g(x) is a reflection of f(x) over the y-axis followed by a translation of 3 units down. Yes, g(x) is a reflection of f(x) over the x-axis followed by a translation of 3 units down.
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How do you calculate (1+i)/(1−i)?
Answer:
i
Step-by-step explanation:
1. Multiply the Numerator and the Denominator by the Complex Conjugate of the Denominator:
[tex]\frac{1+i}{1-i}*\frac{1+i}{1+i}=\frac{1+2i+i^{2}}{1-i^{2}}[/tex]
Remember that i² = -1:
[tex]\frac{1+2i+i^{2}}{1-i^{2}}=\frac{2i}{2}[/tex]
2. Simplify
a jar contains 4 red, 7 white and 10 blue marbles. a child selects two marbles at random from the jar. find the probability that the first is red and the second is white.
The probability that the first is red and the second is white is 4/21 or 0.190476.
The total number of marbles in the jar is 4 red, 7 white and 10 blue. This means there is a total of 21 marbles.
The probability of selecting a red marble first is 4/21 (4 out of 21 marbles are red).
The probability of selecting a white marble second is 7/20 (7 out of the remaining 20 marbles are white).
The probability of selecting a red marble first and a white marble second is 4/21 x 7/20 = 28/420 = 4/60 = 0.190476
The probability of selecting two marbles at random from the jar is calculated by multiplying the individual probabilities of selecting each marble. The probability of selecting the first marble is 4/21 (4 out of 21 marbles are red) and the probability of selecting the second marble is 7/20 (7 out of the remaining 20 marbles are white). Thus, the probability of selecting a red marble first and a white marble second is 4/21 x 7/20 = 28/420 = 4/60 = 0.190476.
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jenn switched jobs to one that was within a half mile of her home and she could walk on most days. after about three months, her new boss announced the company was moving 15 miles outside the city. this would mean that jenn would now have an hour and half commute that involved a subway and two busses. her stressor would be considered a
A dictator
Step-by-step explanation:
quota sampling occurs when available or accessible people or materials are selected, and the sample is weighted so that certain predetermined characteristics are represented. group of answer choices true false
Following the Sampling Theorem of statistics, the answer is True.
What are statistics?The study of statistics focuses on gathering, organizing, organizing, analyzing, interpreting, and presenting data.
What is Sampling Theorem?Each member of the group or community has an equal chance of being chosen when using probability sampling. According to probability-based statistical theory, this indicates that the sample is more likely to match the larger population, allowing for the drawing of more precise conclusions about the latter.
Yes, the quota sampling process occurs when the available or accessible people or material are selected. Due to large population size, Sampling process is preferred.
Since, Statistics convey characteristics of data to us. So yes, The sample is weighted so that predetermined characteristics are represented.
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Sport Hours
Soccer 5
jogging 20 what percentage of total hours was spent skiing
walking 10
skiing 5
football 10
total 50
The percentage of total hours was spent skiing is 10%
What is percentage?Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol "%".
Here, total number of hours =5+20+10+5+10 =50
Number of spent on skiing is 5
Now, percentage is 5/50 ×100
=10%
Therefore, percentage of hours spent on skiing is 10%
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"Your question is incomplete, probably the complete question/missing part is:"
What percentage of total hours were spent skiing?
Soccer 5
Jogging 20
walking 10
skiing 5
football 10
using a survey, we have obtained a large dataset of how many close friends people have. its mean is 6 and its median is 4. the variance is 64. we calculate the standard scores. what is the value of the median of the standardized dataset?
The standard median value of the data set is -0.25.
The standard value of any data is the converted value of the variable, such that the mean of the data set is 0 and the standard deviation is 1.
To convert any data set to its standard form we use the equation
(X - μ)/σ
where,
μ = mean
σ = standard deviation
here,
μ = 6
σ = √64 = 8
Now median is that value which gives the midpoint of the frequencies.
Hence the standard median should be the standard value of the median obtained from the data set which is 4
Hence standard median
= (4 - 6)/8
= -2/8
= -0.25
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Which one of the following is a zero of the polynomial 3x3+4x2−7 ?A. 0
B. 1
C. 2
D. −1
We know that the required zero of the given polynomial 3x²+4x²-7=0 is (B) x = 1.
What are polynomials?A polynomial is a mathematical formula that involves only the operations of addition, subtraction, multiplication, and powers of positive integers of the variables and is composed of coefficients and indeterminates.
An example of a polynomial with a single indeterminate x is x² - 4x + 7.
So, we have the equation:
3x²+4x²-7=0
Now, calculate the zero of the polynomial as follows:
When x = 0:
P(x) = 3x² + 4x² - 7 = 0
P(0) = 3(0)² + 4(0)² - 7
P(0) = - 7
Not zero
When x = 1:
P(x) = 3x² + 4x² - 7 = 0
P(1) = 3(1)² + 4(1)² - 7
P(1) = 3 + 4 - 7
P(1) = 0
Zero of the polynomial.
Therefore, we know that the required zero of the given polynomial 3x²+4x²-7=0 is (B) x = 1.
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Send help! Let angle 1 and angle 2 be complementary angles, and let angle 2 and angle 3 be supplementary angles. If angle 1 = x and angle 2 = 2x, determine x. Then find angle 1, angle 2, and angle 3.
Answer:
1. x = 30
2. ∠1 = 30°
3. ∠2 = 60°
4. ∠3 = 120°
Step-by-step explanation:
⭐What are supplementary angles?
two or more angles that add up to 180°⭐What are complementary angles?
two or more angles that add up to 90°Set up the equations for the situation:
∠1 + ∠2 = 90
∠2 + ∠3 = 180
x + 2x = 90 . . . (1)
2x + ∠3 = 180 . . . (2)
First, solve for "x" in equation 1:
x + 2x = 90
3x = 90
∴ x = 30
Next, substitute "x" into ∠1 and ∠2 to solve for ∠1 and ∠2:
∠1 = x
∴ ∠1 = 30°
∠2 = 2x
∠2 = 2(30)
∴ ∠2 = 60°
Finally, substitute "x" into equation 2 to solve for ∠3:
2x + ∠3 = 180
2(30) + ∠3 = 180
60 + ∠3 = 180
∴ ∠3 = 120°
if you wander around central park in new york city, you will eventually encounter random large boulders that are made of a type of rock not found in the area. how did these large boulders get to central park?
The random large boulders around central park in new york city got to central park They were carried by glaciers and dropped as glacial erratics.
There have been four kinds of bedrock in Manhattan, including the Manhattan schist and the Hart-land shirts, which are metamorphosed sedimentary rocks found in outcrops. Some of these are made out of igneous rocks that developed billions of years ago. It happened during the period of the Grenville progeny and occurred during the formation of the ancient mega continent.
Various glaciers have covered the region of Central Park in the past, the much more recent being the Wisconsin glacier that developed about around 12,000 years ago.
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Which of the following is a zero of the polynomial 3x² - 8x + 5 ?*
0
-1
1
2
The zero of the polynomial 3x² - 8x + 5 is 1
The correct answer option is option C.
How to find zero of a Polynomial?3x² - 8x + 5
check all options:
When x = 0
3x² - 8x + 5
substitute x = 0 into the equation
3(0)² - 8(0) + 5
3(0) - 0 + 5
0 - 0 + 5
= 5
When x = -1
3x² - 8x + 5
3(-1)² - 8(-1) + 5
3(1) + 8 + 5
3 + 8 +5
= 16
When x = 1
3x² - 8x + 5
3(1)² - 8(1) + 5
3(1) - 8 + 5
3 - 8 + 5
= 0
Consequently, 1 is the zero of the given Polynomial.
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a non-for-profit consumer agency wanted to investigate the economic and humanitarian contribution of this country's pharmaceutical companies by sampling large companies and businesses of all types. it was decided to sample the pharmaceutical companies first. after surveying the pharmaceutical companies, someone at the consumer agency wanted to distribute the data early. what, if any, will be the most noticeable bias in distributing the survey at this point in time?
Under coverage of the population will be the most noticeable bias in distributing the survey at his point in time.
Undercoverage bias happens when segments of the target population are entirely executed or loses represented in the sample than they are in the population. This means that these segments are executes from the sampling process. Non response bias occurs when parts of the sampled population are unable or refuse to respond.
Undercoverage occurs when an element of the target population is not represented on the survey frame and therefore not given any chance of selection in the survey sample; that is the element has zero probability of selection into the sample. It refers to a type of sampling bias that occurs when a piece of information from your sample responses goes missing or uncovered in the results. This significant entity goes unselected or has zero chance of getting in your representing sample.
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The width of a rectangle is the length minus 3 units. The area of the rectangle is 28
units. What is the length, in units, of the rectangle?
Devon pays $19.95 for her roller skates. After she pays $6.95 for each visit to the roller rink. What is the greatest number of visits she can afford if the total amount she spends cannot be more than $82.50
Answer: 9 visits
Step-by-step explanation:
Let x = number of visits to the rink
19.95 + 6.95x ≤ 82.5
6.95x ≤ 82.50 - 19.95
6.95x ≤ 62.55
x ≤ 62.55/6.95
x≤ 9
Need help on tis promblommm
Answer:
C,D
Step-by-step explanation:
C and D are multiplying 250 by a number greater than one so the result will be greater than 250
The first two are multiplying 250 by a frction less than one....the result will be less than 250
a kite is flying 120 ft above the ground. the length of the string to the kite is 200 feet, measured from the ground. find the angle, to the nearest degree, that the string makes with the ground.\
The angle, to the degree, that the string makes with the ground is
34.33.
A right-angled triangle can be formed by the string, the height of the kite, and the horizontal distance along the ground to a point directly beneath the kite.
The string represents the hypotenuse (200 feet), and the height (120 feet) represents the opposite side.
The trigonometric ratio sin can be used to calculate the angle between the string and the ground.
sinθ= Perpendicular/hypotenuse
sinθ=120/200
=0.6
Now sin(0.6) = 0.5646
sinθ=0.5646
therefore ,θ
=sin⁻¹(0.5646)
=34.33°
Heights and Distances is a study of Trigonometry applications. It has many real-world applications, such as measuring the height of an object, the depth of an object, the distance between two celestial objects, and so on.
Trigonometric formulas and ratios are useful in the study of heights and distances.
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in a survey conducted on an srs of 200 american adults, 72% of them said they believed in aliens. give a 95% confidence interval for percent of american adults who believe in aliens.
A 95% confidence interval for percent of american adults who believe in aliens: (0.6578, 0.7822)
In this question we have been given that in a survey conducted on an srs of 200 american adults, 72% of them said they believed in aliens
We need to find the 95% confidence interval for percent of american adults who believe in aliens.
95% confidence interval = (p ± z√[p(1 - p)/n])
Here, n = 200
p = 72%
p = 0.72
And the z-score for 95% confidence interval is 1.960
The upper limit of interval would be,
(p + z√[p(1 - p)/n])
= 0.72 + 1.960 √[0.72(1 - 0.72)/200]
= 0.72 + 1.960 √[(0.72 * 0.28)/200]
= 0.72 + 1.960 √0.001008
= 0.72 + 0.0622
= 0.7822
The lower limit of interval would be,
(p - z√[p(1 - p)/n])
= 0.72 - 1.960 √[0.72(1 - 0.72)/200]
= 0.72 - 1.960 √[(0.72 * 0.28)/200]
= 0.72 - 1.960 √0.001008
= 0.72 - 0.0622
= 0.6578
Therefore, a 95% confidence interval = (0.6578, 0.7822)
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how many length m words can be formed from an n-letter alphabet, if no letter is used more than once? (b) how many length m words can be formed from an n-letter alphabet, if letters can be reused? (c) how many binary relations are there from set a to set b when jaj d m and jbj d n? (d) how many total injective functions are there from set a to set b, where jaj d m and jbj d n m
As per the combination method, there are 120, 5 length m words can be formed from an n-letter alphabet, if no letter is used more than once.
Combination method:
in statistics, mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter is known as combination method.
Given,
Here we need to find how many length m words can be formed from an n-letter alphabet, if no letter is used more than once.
Let us consider letters (a,b,c,d,e) how many distinct n length words can be formed.
Here we note that every word can contain same letter m times at most.
So, it can be calculated as,
=> 5 x 4 x 3 x 2 x 1
=> 120
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HELP MEEEEEEEEEEEEEEEEEEEEEEEE PLEASEEEEEEEEEEEEEEEEEEEEE!!!!!!!!!!!!!!!!!!!!!!!
10 coupons are given to 20 shoppers in a store. each shopper can receive at most one coupon. 5 of the shoppers are women and 15 of the shoppers are men. (a) if the coupons are identical, how many ways are there to distribute the coupons so that at least one woman receives a coupon? (b) if the coupons are different, how many ways are there to distribute the coupons so that at least one woman receives a coupon? (c) if the coupons are identical, how many ways are there to distribute the coupons so that at least one woman does not receive a coupon? (d) if the coupons are different, how many ways are there to distribute the coupons so that at least one woman does not receive a coupon?
By adding up all the possible ways to distribute the coupon and subtracting it from the possible ways to distribute the 10 coupons to the 15 men, we can determine the total number of ways at least a woman can receive a coupon.
- The entire number of options for dividing the coupons among the 20 customers is the same as the total number of ways to choose a subset of 10 items from a set of 20.
In other terms, 20/10 = 20!/ 10!10 is the combinatorial number of 20 and 10.
- We must use the same calculation as above, but with a set of 15 items rather than 20, to get the total number of alternatives for distributing the coupons among the 15 males.
We now have 15/10, which is 15!/10!5! possibilities.
To ensure that at least one woman receives a coupon, we have 20/10 - 15/10 distribution options.
What does "combination" mean?A combination is made up of r items chosen without replacement from a set of n items, where the order is irrelevant.
How many different ways that n other items can be combined in a circle exist?The amount of circular permutations of n other objects, taken r at a time, is nPr 2r if clockwise and anti-clockwise configurations are not considered to be distinct. Combinations are various groups or choices made from a variety of items, either all or some of them.
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