select the correct answer.\njudges at an art competition must select first-, second-, and third-place…

select the correct answer.\njudges 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?\na. 1,320\nb. 440\nc. 220\nd. 18

select the correct answer.\njudges 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?\na. 1,320\nb. 440\nc. 220\nd. 18

Answer

Explanation:

Step1: Identify the problem type

This is a permutation problem since the order (first, second, third place) matters. The formula for permutations is ( P(n, r)=\frac{n!}{(n - r)!} ), where ( n = 12 ) (total paintings) and ( r = 3 ) (places to choose).

Step2: Apply the permutation formula

Substitute ( n = 12 ) and ( r = 3 ) into the formula: [ \begin{align*} P(12, 3)&=\frac{12!}{(12 - 3)!}\ &=\frac{12!}{9!}\ &=\frac{12\times11\times10\times9!}{9!}\ &=12\times11\times10\ & = 1320 \end{align*} ]

Answer:

A. 1,320