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?
Answer Choices:
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Solution:
Number the spider's legs from 1 through 8 , and let and denote the sock and shoe that will go on leg . A possible arrangement of the socks and shoes is a permutation of the sixteen symbols , in which precedes for . There are 16 ! permutations of the sixteen symbols, and precedes in exactly half of these, or permutations. Similarly, precedes in exactly half of those, or permutations. Continuing, we can conclude that precedes for in exactly permutations.
The problems on this page are the property of the MAA's American Mathematics Competitions