# 2001 AMC 12 Problems/Problem 16

## Problem

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? $\text{(A) }8! \qquad \text{(B) }2^8 \cdot 8! \qquad \text{(C) }(8!)^2 \qquad \text{(D) }\frac {16!}{2^8} \qquad \text{(E) }16!$

## Best Solution

Pretty clearly, if we choose each sock-shoe pair individually, they will automatically align themselves. $\binom{16}{2}\binom{14}{2}\cdots\binom{2}{2} = \frac{16 \cdot 15 \cdots 2}{2 \cdot 2 \cdots 2} = \frac{16!}{2^8}$

## Solution

### Solution 1

Let the spider try to put on all $16$ things in a random order. Each of the $16!$ permutations is equally probable. For any fixed leg, the probability that he will first put on the sock and only then the shoe is clearly $\frac{1}{2}$. Then the probability that he will correctly put things on all legs is $\frac{1}{2^{8}}$. Therefore the number of correct permutations must be $\boxed{\frac {16!}{2^8}}$.

### Solution 2

Each dressing sequence can be uniquely described by a sequence containing two $1$s, two $2$s, ..., and two $8$s -- the first occurrence of number $x$ means that the spider puts the sock onto leg $x$, the second occurrence of $x$ means he puts the shoe onto leg $x$. If the numbers were all unique, the answer would be $16!$. However, since 8 terms appear twice, the answer is $\frac{16!}{(2!)^8} = \boxed{\dfrac {16!}{2^8}}$.

### Solution 3 (Bashy Solution)

You can put all $8$ socks on first for $8!$ ways and then all $8$ shoes on next for $8!$ more ways. This is not the only possibility, so the lower bound is $(8!)^2$. You can choose all $16$ in a random fashion, but some combinations would violate the rules, so the upper bound is $16!$. $\text{(C)}$ & $\text{(E)}$ are the lower and upper bounds, so the answer is in between them, $\boxed{\frac {16!}{2^8}}$.

## Also See

 2001 AMC 12 (Problems • Answer Key • Resources) Preceded byProblem 15 Followed byProblem 17 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

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