Question

select the correct answer.
judges at an art competition must select first-, second-, and third-place winners from an exhibition of 12 paintings. in how many different ways can the winning paintings be chosen?
a. 1,320
b. 440
c. 220
d. 18

Explanation:

Step1: Identify permutation need

Since order (1st, 2nd, 3rd) matters, we use permutations. The formula for permutations of $n$ items taken $k$ at a time is $P(n,k)=\frac{n!}{(n-k)!}$

Step2: Plug in values

Here $n=12$, $k=3$.
$$P(12,3)=\frac{12!}{(12-3)!}=\frac{12!}{9!}$$

Step3: Simplify the expression

$12! = 12\times11\times10\times9!$, so $\frac{12\times11\times10\times9!}{9!}=12\times11\times10$

Step4: Calculate final value

$12\times11\times10=1320$

Answer:

A. 1,320