Problem:
Melinda has three empty boxes and textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as , where and are relatively prime positive integers. Find .
Solution:
The probability that all three mathematics textbooks end up in the first box is the probability that the three mathematics textbooks are selected out of the equally likely ways to select three textbooks. That is . The probability that all three mathematics textbooks end up in the second box is the probability that three mathematics textbooks and one other of the nine remaining textbooks are selected for the second box out of the equally likely ways to select four textbooks. That is . Finally, the probability that all three mathematics textbooks end up in the last box is . The probability that all three mathematics textbooks end up in the same box is . The requested sum is .
Let . There are ways for Melinda to place her books in the boxes. There are ways with all the math books in the first box, ways with all of them in the second box, and ways with all of them in the third box. Thus the probability is