Combinations question

stephkhaira wrote:Codes can be formed by 1, 2 or 3 letters from the 26-letter alphabet. If letters can be repeated and if the same letters in different order are taken as different codes, then how many codes can be formed?

A) 2951
B) 8125
C) 15600
D) 16302
E) 18278


[spoiler]ans: 18278[/spoiler]

Number of codes with 1 letter
26

Number of codes with 2 letters
26 x 26
= 26^2

Number of codes with 3 letters
26 x 26 x 26
= 26^3

IMPORTANT: we do NOT want to calculate 26 + (26^2) + (26^3)
The GMAT test-makers would never require us to perform these kinds of lengthy calculations, BUT it's still possible that you could be given answer choices that SUGGEST that we need to actually perform the calculations.

The answer choice here are great in that they might lead you start calculating.
However, if we focus on the units of each part of the calculation, we see a faster way to arrive at the correct answer.

26 has 6 as its units digit
26^2 has 6 as its units digit
26^3 has 6 as its units digit

So, (26)+(26^2)+(26^3) = (26)+(___6)+(____6) = _____8

Since only E has 8 as its units digit, the answer must be E

Cheers,
Brent