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# A certain stock exchange designates each stock with a 1, 2

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A certain stock exchange designates each stock with a 1, 2 [#permalink]  10 Nov 2009, 14:50
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 31 Jul 2012, 12:51, edited 2 times in total.
Edited the question and added the OA
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Re: Permutation Problem [#permalink]  10 Nov 2009, 14:57
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chicagocubsrule wrote:
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

if each letter is the same: 26 different combinations
2 letters the same 26^2
all different 26^3

26^3 + 26^2 + 26 = 18278
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A certain stock exchange designates each stock with a 1, 2 [#permalink]  10 Nov 2009, 14:59
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chicagocubsrule wrote:
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

1 letter codes = 26
2 letter codes = 26^2
3 letter codes = 26^3

Total = 26 + 26^2 + 26^3

The problem we are faced now is how to get the answer quickly. Note that the units digit of 26+26^2+26^3 would be (6+6+6=18) 8. Only one answer choice has 8 as unit digit: E (18,278). So I believe, even not calculating 26+26^2+26^3, that answer is E.
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Re: Stock exchange [#permalink]  01 Nov 2010, 18:19
The number of combinations for a stock w/ one letter is simply 26 (26 letters).

The number of combinations for a stock w/ two letters is 26*26 = 676.

The number of combinations for a stock w/ three letters is 26*26*26 = 17576.

Summing all of the possible combinations results in 17576 + 676 + 26 = 18278, hence answer E.
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Re: Permutation Problem [#permalink]  31 Jan 2011, 08:48
Bunuel wrote:
chicagocubsrule wrote:
The problem we are faced now is how to get the answer quickly. Note that the units digit of 26+26^2+26^3 would be (6+6+6=18) 8. Only one answer choice has 8 as unit digit: E (18,278). So I believe, even not calculating 26+26^2+26^3, that answer is E.

The OA is E. Thanks for poininting out how to spot the correct answer - it took me miserable 4 minutes to multiply 26*26*26 and still I made a wrong calculation
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Re: Permutation Problem [#permalink]  14 Feb 2013, 08:18
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Bunuel wrote:
chicagocubsrule wrote:
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

1 letter code=26
2 letter code=26^2
3 letter code=26^3

Total=26+26^2+26^3

The problem we are faced now is how to get the answer quickly. Note that the units digit of 26+26^2+26^3 would be (6+6+6=18) 8. Only one answer choice has 8 as unit digit: E (18,278). So I believe, even not calculating 26+26^2+26^3, that answer is E.

Hi Bunuel,

Firstly let me say that i fully understand your explanation and it makes perfect sense. I am however, finding it difficult to understand why we can't plug in the numbers into the permutations formula i.e. 26+Pm26,2 + Pm26,3 =16,276 which is well short of the 18,278 answer. I'm just wondering when to apply the approach you mentioned above and when to apply the Permutations formula.

Thanks!
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Re: Permutation Problem [#permalink]  14 Feb 2013, 09:25
26 + 26^2 + 26^3 = 26+676+17576=18278
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Re: Permutation Problem [#permalink]  15 Feb 2013, 02:51
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iwillbeatthegmat wrote:
Bunuel wrote:
chicagocubsrule wrote:
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

1 letter code=26
2 letter code=26^2
3 letter code=26^3

Total=26+26^2+26^3

The problem we are faced now is how to get the answer quickly. Note that the units digit of 26+26^2+26^3 would be (6+6+6=18) 8. Only one answer choice has 8 as unit digit: E (18,278). So I believe, even not calculating 26+26^2+26^3, that answer is E.

Hi Bunuel,

Firstly let me say that i fully understand your explanation and it makes perfect sense. I am however, finding it difficult to understand why we can't plug in the numbers into the permutations formula i.e. 26+Pm26,2 + Pm26,3 =16,276 which is well short of the 18,278 answer. I'm just wondering when to apply the approach you mentioned above and when to apply the Permutations formula.

Thanks!

Good question. +1.

Notice that we are told that the letters may be repeated, so AA, BBB, ACC, CAA, .... codes are possible.

Now, 26P2 is the number of ways we can choose 2 distinct letters out of 26 when the order matters, thus it doesn't account for the cases like AA, AAA, ABB, ...

Hope it's clear.
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]  15 Feb 2013, 03:04
Perfectly clear! The repetition disqualifies the permutations formula.

Thanks alot Bunuel!
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A certain stock exchange designates each stock with a 1, 2, or 3 [#permalink]  23 Mar 2013, 10:12
Also note that you do not have to multiply everything out - just look at the UNITS DIGIT for each number that you multiply/add

26*26*26 = ONES DIGIT IS 6
26*26 = ONES DIGIT IS 6
26 = ONES DIGIT IS 6

The ones digit of the final answer will be 6 + 6 + 6 ... which is 18

The only answer with an 8 in the ones digit is E
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Re: Permutation Problem [#permalink]  18 Sep 2013, 02:56
lagomez wrote:
chicagocubsrule wrote:
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

if each letter is the same: 26 different combinations
2 letters the same 26^2
all different 26^3

26^3 + 26^2 + 26 = 18278

what does this statement exactly mean-
"if the same letters used in a different order constitute a different code"
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Re: Permutation Problem [#permalink]  18 Sep 2013, 03:08
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honchos wrote:
lagomez wrote:
chicagocubsrule wrote:
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

if each letter is the same: 26 different combinations
2 letters the same 26^2
all different 26^3

26^3 + 26^2 + 26 = 18278

what does this statement exactly mean-
"if the same letters used in a different order constitute a different code"

It means that the order of the letters matters. For example, code AB is different from BA.

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Re: Permutation Problem [#permalink]  18 Jan 2014, 05:48
iwillbeatthegmat wrote:
Bunuel wrote:
chicagocubsrule wrote:
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

1 letter code=26
2 letter code=26^2
3 letter code=26^3

Total=26+26^2+26^3

The problem we are faced now is how to get the answer quickly. Note that the units digit of 26+26^2+26^3 would be (6+6+6=18) 8. Only one answer choice has 8 as unit digit: E (18,278). So I believe, even not calculating 26+26^2+26^3, that answer is E.

Hi Bunuel,

Firstly let me say that i fully understand your explanation and it makes perfect sense. I am however, finding it difficult to understand why we can't plug in the numbers into the permutations formula i.e. 26+Pm26,2 + Pm26,3 =16,276 which is well short of the 18,278 answer. I'm just wondering when to apply the approach you mentioned above and when to apply the Permutations formula.

Thanks!

1 letter code: 26
2-letter code: P(26,2) + 26 {P(26,2): 2 different numbers and different orders; 26: 2 same numbers}
3-letter code: P(26,3) + P(26, 2)C(3, 1) + 26 {P(26,3): 3 different numbers and different orders; P(26, 2)C(3, 1): 2 different numbers, one of which repeats; 26: 3 same numbers}

Hope it helps to understand.
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]  29 Oct 2014, 00:59
I dont know why , but I was thinking for one letter, it's 26,
Then for 2 same ones it would be 26^2
2 different ones would mean 26*25 * 2 (because a different order)
3 same would be 26^3, and 3 different would be 26*25*24*3!....Where am I (obviously) double counting?
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]  29 Oct 2014, 02:48
Expert's post
usre123 wrote:
I dont know why , but I was thinking for one letter, it's 26,
Then for 2 same ones it would be 26^2
2 different ones would mean 26*25 * 2 (because a different order)
3 same would be 26^3, and 3 different would be 26*25*24*3!....Where am I (obviously) double counting?

How is 26^2 the number of two same letter words? How is 26^3 the number of three same letter words? Isn't both 26? AA, BB, CC, ..., ZZ and AAA, BBB, CCC, DDD, ..., ZZZ?

26^2 gives the number of ALL 2-letter words possible, the same way as 26^3 gives the number of ALL 3-letter words possible.
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]  25 Nov 2014, 04:16
Hi,

I have a doubt. If this question mentioned that same letters used in a different order are considered same code, then how would we solve it?

Thanks
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]  25 Nov 2014, 04:22
Bunuel wrote:
usre123 wrote:
I dont know why , but I was thinking for one letter, it's 26,
Then for 2 same ones it would be 26^2
2 different ones would mean 26*25 * 2 (because a different order)
3 same would be 26^3, and 3 different would be 26*25*24*3!....Where am I (obviously) double counting?

How is 26^2 the number of two same letter words? How is 26^3 the number of three same letter words? Isn't both 26? AA, BB, CC, ..., ZZ and AAA, BBB, CCC, DDD, ..., ZZZ?

26^2 gives the number of ALL 2-letter words possible, the same way as 26^3 gives the number of ALL 3-letter words possible.

Crystal clear when I read this, thanks!
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]  12 May 2015, 06:26
Bunuel wrote:
chicagocubsrule wrote:
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

1 letter codes = 26
2 letter codes = 26^2
3 letter codes = 26^3

Total = 26 + 26^2 + 26^3

The problem we are faced now is how to get the answer quickly. Note that the units digit of 26+26^2+26^3 would be (6+6+6=18) 8. Only one answer choice has 8 as unit digit: E (18,278). So I believe, even not calculating 26+26^2+26^3, that answer is E.

can not say a word for this excellent
Re: A certain stock exchange designates each stock with a 1, 2   [#permalink] 12 May 2015, 06:26
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