Combination - could not find the answer - please help

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A certain stock exchange designates each stock with a one-, two- or three-letter code, where each letter is selected from the 26 letters of the alphabets. If the letter maybe repeated and if the same letters used in different order constitude a different code, how many different stock is it possible to uniquely designate with these codes?

A. 2,951
B. 8,125
C. 15,600
D. 16,302
E. 18,278

I dont know how to hide the answer so I will post the answer later.

Thank you for your help!

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by Brent@GMATPrepNow » Sat Sep 14, 2013 6:09 am
VyDinh wrote:A certain stock exchange designates each stock with a one-, two- or three-letter code, where each letter is selected from the 26 letters of the alphabets. If the letter maybe repeated and if the same letters used in different order constitude a different code, how many different stock is it possible to uniquely designate with these codes?

A. 2,951
B. 8,125
C. 15,600
D. 16,302
E. 18,278
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) = 26^2

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) = 26^3

So, the TOTAL number of codes = 26 + 26^2 + 26^3

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

So, (26)+(26x26)+(26x26x26) = (26)+(___6)+(____6) = _____8

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

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Sat Apr 05, 2014 5:24 pm, edited 1 time in total.
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by vinay1983 » Sat Sep 14, 2013 6:36 am
Order not important and repetition allowed is the given condition

1 letter code is used, then no.of ways of designating code is 26 only
OR
2 letter codes are used, then 26*26 = 676
OR
3 letter codes are used, then 26*26*26= 17576

Add all the values i.e., 26+676+17576 = 18278

Option E is correct.

P.S: to hide the OA, first type "OA E", then select the word or phrase between the apostrophes eg here (OA E) and then click on the spoiler button. Done!
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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by VyDinh » Sat Sep 14, 2013 7:50 am
Thanks guys. E is the correct answer choice.

I think I misunderstand the question. I think they are asking that each stock is a 3-letter codes so I missed the 1-letter and 2-letters stock.