how many ways can a high school basketball coach find the 5 players for her starting lineup from 14 players?

Explanation:

Step1: Determine the formula

This is a permutation problem. The formula for permutations is (P(n,r)=\frac{n!}{(n - r)!}), where (n) is the total number of items, and (r) is the number of items to be selected. Here, (n = 14) and (r=5).

Step2: Calculate factorial values

First, calculate (n!=14! = 14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1) and ((n - r)!=(14 - 5)!=9!=9\times8\times7\times6\times5\times4\times3\times2\times1). Then (P(14,5)=\frac{14!}{9!}=\frac{14\times13\times12\times11\times10\times9!}{9!}).

Step3: Simplify the expression

Cancel out the (9!) terms. So (P(14,5)=14\times13\times12\times11\times10). [ \begin{align*} 14\times13\times12\times11\times10&=(14\times13)\times12\times11\times10\ &=182\times12\times11\times10\ &=2184\times11\times10\ &=24024\times10\ &=240240 \end{align*} ]

Answer:

240240