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A company assigns product codes consisting of all the letter

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A company assigns product codes consisting of all the letter  [#permalink]

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01 Sep 2013, 07:49
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Difficulty:

65% (hard)

Question Stats:

55% (01:41) correct 45% (01:44) wrong based on 201 sessions

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A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

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Re: A company assigns product codes consisting of all the letter  [#permalink]

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06 Feb 2016, 15:49
5
bagdbmba wrote:
A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

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 D has 8 as its units digit, the answer must be D

Cheers,
Brent
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Re: A company assigns product codes consisting of all the letter  [#permalink]

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01 Sep 2013, 08:20
7
bagdbmba wrote:
A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

The no of ways in which the company can make codes of 1 letter : 26
# for 2 letters : 26*26 = _ _ 6
# for 3 letters : _ _ 6 * 26 =_ _ _ _ 6

The total : By adding all of them , the units digit would be : 6+6+6 = _ _ _ _ _ 8

D.
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Re: A company assigns product codes consisting of all the letter  [#permalink]

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01 Sep 2013, 10:58
1
bagdbmba wrote:
A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

Similar question from GMAT Prep:
Quote:
A certain stock exchange designates each stock with a 1, 2, or 3 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. 2,951
B. 8,125
C. 15,600
D. 16,302
E. 18,278

Discussed here: a-certain-stock-exchange-designates-each-stock-with-a-86656.html

Other similar questions to practice:
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Hope it helps.
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Re: A company assigns product codes consisting of all the letter  [#permalink]

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03 Sep 2013, 08:28
1
Thanks Bunuel for not only referring to the solution but also sharing several similar problems for practice.
Really appreciate it.

+1 Sir!
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Re: A company assigns product codes consisting of all the letter  [#permalink]

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06 Feb 2016, 14:33
mau5 wrote:
bagdbmba wrote:
A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

The no of ways in which the company can make codes of 1 letter : 26
# for 2 letters : 26*26 = _ _ 6
# for 3 letters : _ _ 6 * 26 =_ _ _ _ 6

The total : By adding all of them , the units digit would be : 6+6+6 = _ _ _ _ _ 8

D.

How could I use the combinatorics formula to solve this? I'm not getting the correct answer if I try the formula: nCk= n!/k!(n-k)!
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Re: A company assigns product codes consisting of all the letter  [#permalink]

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05 Jun 2017, 04:12
Bunuel

Isn't there double counting if we do 26 * 26 * 26?
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Re: A company assigns product codes consisting of all the letter  [#permalink]

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06 Jun 2017, 16:38
bagdbmba wrote:
A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

A one-letter code can be formed in 26 ways.

A two-letter code can be formed in 26 x 26 = 26^2 ways.

A three-letter code can be formed in 26 x 26 x 26 = 26^3 ways.

Thus, the total number of codes is:

26 + 26^2 + 26^3

26(1 + 26 + 26^2)

26(1 + 26 + 676)

26(703) = 18,278

Notice that one need not do the multiplication out fully. We see that the units digit is an 8.

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Re: A company assigns product codes consisting of all the letter  [#permalink]

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14 Aug 2017, 02:54
bagdbmba wrote:
A company assigns product codes consisting of all the letters in the alphabet.How many product codes are possible if the company uses at most 3 letters in its codes, and all letters can be repeated in any one code?

A.15600
B.16226
C.17576
D.18278
E.28572

So product codes can be o f 1 letter or 2 letter or 3 letters and all the letters can be repeated in any code.
So product codes possible = 26 + 26*26 + 26*26*26 = 26(1+26+26^2) = 26*(27+676) = 26* (703) = 18278

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Re: A company assigns product codes consisting of all the letter  [#permalink]

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08 Apr 2019, 07:58
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Re: A company assigns product codes consisting of all the letter   [#permalink] 08 Apr 2019, 07:58
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