Problem:
Ellina has twelve blocks, two each of red (R), blue (B), yellow (Y), green (G), orange (O), and purple (P). Call an arrangement of blocks even if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is , where and are relatively prime positive integers. Find .
Solution:
Answer (247):
More generally, let there be colors, numbered , with two blocks of each color. Note that an arrangement of blocks is even if and only if every pair of blocks of the same color are in positions whose position numbers differ by an odd number, which means that one position number is even and the other is odd. Thus Ellina forms an even arrangement if and only if there are two permutations of the colors and such that Ellina arranges the blocks in the order . It follows that there is a total of even arrangements. The number of ways to arrange pairs of blocks is . For , the probability that a random arrangement is even equals
The requested sum is .
Note: The number of even arrangements is sequence A001044 in the On-Line Encyclopedia of Integer Sequences.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions