GMATPrep Math Question

[tpetel]

If there is a stock exchange that can use 1,2, or 3 letters to create stock symbols out of the 26 letters in the a-z: How many unique symbols can be formed?

The answer is 18,278.

My initial thought was 26nCr1 + 26nCr2 + 26nCr3

Can someone help me with the solution? Thanks

[DanaJ]

You have 1, 2 or 3 letters. There are 3 possibilities:
a. your symbol is made up of only one letter, making 26 (since you have 26 letters) possibilities.
b. your symbol is made up of two letters. The symbols have to be distinct, but the letters used to make them do not. Therefore, "aa" and "zz" can be counted and the order matters (i.e. "ab" is not the same as "ba"). This makes for 26*26 more symbols, or 676 symbols made of 2 letters.
c. your symbol is made up of two letters, case when you get 26*26*26 = 17 576 possible symbols, since the order does matter and you can use double or triple letters (meaning "aab" or "mmm" are valid combinations).

In total you get 26 + 676 + 17 576 = 18 278 symbols.