Question

there are 14 dogs in a particular class at a dog show. blue, red, yellow, and white ribbons will be awarded to the first through fourth place finishers, respectively. in how many different ways can the ribbons be awarded?
1,001
6,006
24,024
38,416

Explanation:

Step1: Identify permutation formula

We use permutations since order matters (1st-4th place are distinct). The formula for permutations is $P(n,k)=\frac{n!}{(n-k)!}$, where $n=14$ (total dogs) and $k=4$ (ribbons to award).

Step2: Substitute values into formula

$$P(14,4)=\frac{14!}{(14-4)!}=\frac{14!}{10!}$$

Step3: Simplify the expression

$\frac{14!}{10!}=14\times13\times12\times11$

Step4: Calculate the product

$14\times13=182$, $182\times12=2184$, $2184\times11=24024$

Answer:

24,024