swerve wrote:A certain exchange designates each stock with one-, two-, or three-letter code, where each letter is selected from the 26 letters of the alphabet. If the letters may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate these codes?
A. 2951
B. 8125
C. 15600
D. 16302
E. 18278
The OA is E
Source: GMAT Prep
1-letter codes
26 letters, so there are 26 possible codes
2-letter codes
There are 26 options for the 1st letter, and 26 options for the 2nd letter.
So, the number of 2-letter codes = (26)(26) = 26²
3-letter codes
There are 26 options for the 1st letter, 26 options for the 2nd letter, and 26 options for the 3rd letter.
So, the number of 3-letter codes = (26)(26)(26) = 26³
So, the TOTAL number of codes = 26 + 26² + 26³
IMPORTANT: Before we perform ANY calculations, we should first look at the answer choices, because we know that the GMAT test-makers are very reasonable, and they don't care whether we're able make long, tedious calculations. Instead, the test-makers will create the question (or answer choices) so that there's an alternative approach.
The alternative approach here is to recognize that:
26 has
6 as its units digit
26² has
6 as its units digit
26³ has
6 as its units digit
So, (26)+(26²)+(26³) = (2
6)+(___
6)+(____
6) = _____
8
Since only E has
8 as its units digit, the answer must be E
Cheers,
Brent
