stock exchange-color code

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[email protected]: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Tue Apr 02, 2013 7:42 am

by [email protected] » Sun May 12, 2013 11:02 am

A certain stock exchange designates each stock with the 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 with these codes?

a)2951
b)8125
c)15600
d)16302
e)18278

OA is E

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by GMATGuruNY » Sun May 12, 2013 11:38 am

[email protected] wrote:A certain stock exchange designates each stock with the 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 with these codes?

a)2951
b)8125
c)15600
d)16302
e)18278

OA is E

One-letter codes:
Number of options = 26.

2-letter codes:
Number of options for the first letter = 26.
Number of options for the second letter = 26.
To combine these options, we multiply:
26*26.

3-letter codes:
Number of options for the first letter = 26.
Number of options for the second letter = 26.
Number of options for the third letter = 26.
To combine these options, we multiply:
26*26*26.

Total possible codes = 26 + (26*26) + (26*26*26).
Each answer choice has a different units digit, so we can determine the correct answer simply by identifying the units digit of the sum above.
26 + (26*26) + (26*26*26) --> (unit digit of 6) + (units digit of 6) + (units digit of 6).
Since 6+6+6 = 18, the correct answer choice must have a units digit of 8.

The correct answer is E.

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Re: stock exchange-color code

by Brent@GMATPrepNow » Sun Apr 03, 2022 6:11 am

[email protected] wrote: ↑
Sun May 12, 2013 11:02 am
A certain stock exchange designates each stock with the 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 with these codes?

a)2951
b)8125
c)15600
d)16302
e)18278

OA is E

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) = (26)+(___6)+(____6) = _____8

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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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