2015 AIME I Problems/Problem 5
Problem
In a drawer Sandy has 
pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 
socks in the drawer. On Tuesday Sandy selects 
of the remaining 
socks at random and on Wednesday two of the remaining 
socks at random. The probability that Wednesday is the first day Sandy selects matching socks is 
, where 
and 
are relatively prime positive integers, Find 
.
Solution 1
Let the fifth sock be arbitrary; the probability that the sixth sock matches in color is 
.
Assuming this, then let the first sock be arbitrary; the probability that the second sock does not match is 
The only "hard" part is the third and fourth sock. But that is simple casework. If the third sock's color matches the color of one of the first two socks (which occurs with probability 
), then the fourth sock can be arbitrary. Otherwise (with probability 
\dfrac{4}{5}
26+315=\boxed{341}.
5
4
\dbinom{8}{2}-4
6
\dbinom{6}{2}-2
26 + 315 = \boxed{341}$.
See also
| 2015 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
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