Question

inspecting restaurants how many different ways can a city health department inspector visit 6 restaurants in a city with 14 restaurants? the order of selection is important. the number of different ways the inspector can visit 6 out of 14 restaurants is \boxed{}

Explanation:

Step1: Identify the problem type

This is a permutation problem since the order of selection is important. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 14 \) (total restaurants) and \( r = 6 \) (restaurants to visit).

Step2: Substitute values into the formula

First, calculate \( n! = 14! \), \( (n - r)!=(14 - 6)!=8! \). So \( P(14, 6)=\frac{14!}{8!} \).

Step3: Simplify the factorials

We know that \( n!=n\times(n - 1)\times\cdots\times(n - r + 1)\times(n - r)! \), so \( \frac{14!}{8!}=14\times13\times12\times11\times10\times9 \).

Step4: Calculate the product

\( 14\times13 = 182 \), \( 182\times12 = 2184 \), \( 2184\times11 = 24024 \), \( 24024\times10 = 240240 \), \( 240240\times9 = 2162160 \).

Answer:

2162160