2001 AMC 12 Problem 16

Question

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

8!

2^8\, 8!

(8!)^2

\dfrac{16!}{2^8}

16!

Answer: D

Concepts:arrangements with restrictions permutations

Difficulty rating: 1600

Small hint:

There are

16

items, so

16!

orderings with no restriction

Big hint:

On each leg the sock precedes the shoe in exactly half of all orderings, and the eight legs are independent

Solution:

Think of the

16

items (

8

socks and

8

shoes) arranged in some order: there are

16!

arrangements.

For each leg, the sock comes before the shoe in exactly half of all arrangements. Imposing this on all eight legs independently divides by

2^8,

giving

\dfrac{16!}{2^8}.

Thus, the correct answer is D.