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Chapter 10: Problem 3

Evaluate each number of permutations or combinations without using your calculator. Show your calculations. a. \({ }_{5} P_{3}\) (a) b. \(5 C_{3}\) c. \(5 P_{4}\) d. \({ }_{5} C_{4}\)

Short Answer

Expert verified
a) 60, b) 10, c) 120, d) 5.

Step by step solution

01

Understanding Permutations and Combinations

Permutations and combinations are different ways of selecting items from a set. A permutation considers the order of items, while a combination does not. We will solve each part using the appropriate permutation or combination formula.
02

Evaluate \( ^{5}P_{3} \)

Permutations of choosing 3 items from 5, where order matters, is calculated as:\[^{5}P_{3} = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 60.\]
03

Evaluate \(5 C_{3}\)

Combinations of choosing 3 items from 5, where order does not matter, use the formula:\[^{5}C_{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} = 10.\]
04

Evaluate \( ^{5}P_{4} \)

Permutations of choosing 4 items from 5, where order matters, is given by:\[^{5}P_{4} = \frac{5!}{(5-4)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{1} = 120.\]
05

Evaluate \( ^{5}C_{4} \)

Combinations of choosing 4 items from 5, where order doesn't matter, is calculated as:\[^{5}C_{4} = \frac{5!}{4!(5-4)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1 \times 1} = 5.\]
06

Final Answer Collection

We can now compile the results:- \( ^{5}P_{3} = 60 \)- \( ^{5}C_{3} = 10 \)- \( ^{5}P_{4} = 120 \)- \( ^{5}C_{4} = 5 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations
Permutations are arrangements of objects where the order is essential. This differs from combinations, where order does not matter. Permutations are often used in situations where we are concerned about the sequence or arrangement of certain items.
For example, if you have five books and want to arrange three of them in a specific sequence, you would use permutations.
To compute permutations of selecting and ordering r items from a set of n, we use the formula:
  • \[ ^n P_r = \frac{n!}{(n-r)!} \]
This formula simplifies to multiplying the first r consecutive numbers of n.
For instance, the permutation \( ^5 P_3 \) is calculated as \( \frac{5!}{(5-3)!} = 5 \times 4 \times 3 = 60\). This means there are 60 different ways to arrange 3 books out of 5.
Combinations
Combinations involve selections where the order of items does not matter. This is useful in scenarios where the sequence or arrangement is irrelevant, only the selection is.
For example, if you have 5 different fruits and want to choose 3 for a fruit salad, combinations would be appropriate.
The formula to find the number of combinations when selecting r items from n is:
  • \[ ^n C_r = \frac{n!}{r!(n-r)!} \]
This formula accounts for all possible selections by dividing through by the redundant arrangements.
As an example, to find the number of ways to choose 3 fruits out of 5, we calculate \( ^5 C_3 \) using the formula: \( \frac{5!}{3!(5-3)!} = 10 \). So, there are 10 different combinations of picking 3 fruits.
Factorial
The factorial function, denoted by an exclamation mark (!), is a core concept used in permutations and combinations. A factorial of a positive integer n is the product of all positive integers less than or equal to n.
For example, the factorial of 5 (written as 5!) is the product \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials are fundamental in determining the total number of permutations and combinations. They simplify how we count large sets and ensure accurate calculations by considering every possible arrangement of the set.
  • They are helpful in organizing and simplifying mathematical problems that involve counting and arranging items.
  • Factorials are easily incorporated into formulas for permutations and combinations to help calculate totals efficiently.
Understanding how to manipulate and apply factorials is key for solving problems in combinatorics, making it an essential skill in statistics and probability.

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Most popular questions from this chapter

Last month it was estimated that a lake contained 3500 rainbow trout. Over a three-day period a park ranger caught, tagged, and released 100 fish. Then, after allowing two weeks for random mixing, she caught 100 more rainbow trout and found that 3 of them had tags. a. What is the probability of catching a tagged trout? b. What assumptions must you make to answer \(2 \mathrm{a}\) ? c. Based on the number of tagged fish she caught two weeks later, what is the park ranger's experimental probability?

For \(10 a-f\), if the number has an exponent, write it in standard form. If the number is in standard form, write it with an exponent other than \(1 .\) a. \(4^{3}\) b. \(\left(\frac{1}{6}\right)^{2}\) c. \(\left(\frac{3}{4}\right)^{2}\) d. 27 c. \(\frac{1}{125}\) f. \(\frac{4}{81}\)

In April 2004 , the faculty at Princeton University voted that each department could give A grades to no more than \(35 \%\) of their students. Japanese teacher Kyoko Loetscher felt that 11 of her 20 students deserved A's, as they had earned better than \(90 \%\) in the course. However, she could give A's to only \(35 \%\) of her students. How many students is this? Draw two relative frequency circle graphs: one that shows the grades (A's versus non-A's) that Loetscher would like to give and one that shows the grades she is allowed to give. (Newsweek, February 14, 2005, p. 8) (a)

The star hitter on the baseball team at City Community College had a batting average of \(.375\) before the start of a three-game series. (Note: Batting average is calculated by dividing hits by times at bat; sacrifice bunts and walks do not count as times at bat.) During the three games, he came to the plate to bat eleven times. In these eleven plate appearances, he walked twice and had one sacrifice bunt. He either got a hit or struck out in his other plate appearances. If his batting average was the same at the end of the three-game series as at the beginning, how many hits did he get?

A thumbtack can land "point up" or "point down." a. When you drop a thumbtack on a hard surface, do you think the two outcomes will be equally likely? If not, what would you predict for \(P(\) up \()\) ? b. Drop a thumbtack 100 times onto a hard surface, or drop 10 thumbtacks 10 times. Record the frequency of "point up" and "point down." What are your experimental probabilities for the two responses? c. Make a prediction for the probabilities on a softer surface like a towel. Repeat the experiment over a towel. What are your experimental probabilities?
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