If out of letters there are , show that the number of ways of selecting letters out of these is the same as selecting letters out of them. If , show that the number of ways of selecting letters is given by
When you select letters, you leave behind letters. Hence the number of ways of selecting letters is the same as the number of ways of leaving behind letters, so
Let be the number of A’s and B’s respectively in the selection.
(1)
When the minimum value of is (since there are at most C’s) and the maximum is Hence there are values for when When so can have values. Continuing, we have that when …. When
Hence the number of selections in which is
(2)
When can range from 0 to so When so Hence: when …, when
Hence the number of selections in which is
Now if you add the results in (1) and (2), you should hopefully find that simplifies to