Evaluate the expression. *permutation notation* the number of permutations 9 things taken 5 at a time (sub 9)P(sub 5)

Verified step by step guidance

1. Understand the permutation formula: The number of permutations of n things taken r at a time is given by the formula: $P(n,r)=\frac{n!}{(n-r)!}$, where n! is the factorial of n.
2. Identify the values of n and r from the problem: Here, n = 9 and r = 5. This means we are calculating the number of permutations of 9 things taken 5 at a time.
3. Substitute the values of n and r into the formula: $P(9,5)=\frac{9!}{(9-5)!}$.
4. Simplify the denominator: Calculate $(9-5)!$, which is $4!$. The formula now becomes: $\frac{9!}{4}$.
5. Expand and simplify the factorials: Write out $9!$ as $9\times 8\times 7\times 6\times 5\times 4!$, and cancel the $4!$ in the numerator and denominator. This leaves: $9\times 8\times 7\times 6\times 5$.

Key Concepts

Permutations

Permutations refer to the different ways in which a set of items can be arranged or ordered. In mathematics, the number of permutations of 'n' items taken 'r' at a time is calculated using the formula P(n, r) = n! / (n - r)!, where 'n!' denotes the factorial of 'n'. This concept is crucial for understanding how to count arrangements when the order of selection matters.

Factorial

A factorial, denoted as 'n!', is the product of all positive integers up to 'n'. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, particularly in calculating permutations and combinations, as they provide a way to quantify the total arrangements of a set of items.

Combination vs. Permutation

The distinction between combinations and permutations is essential in combinatorial mathematics. While permutations consider the order of selection (e.g., ABC is different from ACB), combinations focus solely on the selection itself, disregarding order (e.g., ABC is the same as ACB). Understanding this difference is vital when determining which formula to apply in problems involving arrangements.