Textbook Question
Evaluate the expression. *permutation notation* the number of permutations 9 things taken 5 at a time (sub 9)P(sub 5)
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10. Combinatorics & Probability
Combinatorics
Problem 4
Textbook Question
Use the formula for nPr to evaluate each expression. 10P4
Verified step by step guidance1
Recall the formula for permutations: \(\displaystyle {}_nP_r = \frac{n!}{(n-r)!}\), where \(n\) is the total number of items and \(r\) is the number of items to arrange.
Identify the values of \(n\) and \(r\) from the problem: here, \(n = 10\) and \(r = 4\).
Substitute these values into the permutation formula: \(\displaystyle {}_{10}P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!}\).
Simplify the factorial expression by expanding \$10!\( only as far as needed to cancel with \)6!$: \(\displaystyle \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!}\).
Cancel the \$6!$ terms and multiply the remaining numbers: \(10 \times 9 \times 8 \times 7\) to find the number of permutations.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Permutation Formula
The permutation formula, denoted as nPr, calculates the number of ways to arrange r objects from a set of n distinct objects where order matters. It is given by nPr = n! / (n - r)!, where '!' denotes factorial.
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Introduction to Permutations
Factorial Function
A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in permutations and combinations to count arrangements.
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Evaluating Permutations
To evaluate a permutation like 10P4, substitute into the formula: 10P4 = 10! / (10 - 4)! = 10! / 6!. Simplify by canceling common factorial terms and compute the product of the remaining factors.
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7:11Introduction to Permutations
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