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:
We choose placements for each color pair, following the constraints.
There are ways to place the red blocks, for the blue blocks, for green, for yellow, for orange, and for purple.
Thus, the number of valid arrangements is:
The total number of ways to arrange the 12 blocks without restriction is .
So the desired probability is:
We compute the number of valid arrangements:
The denominator is , so the probability is:
and the final answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions