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A certain stock exchange designates each stock with a 1, 2 [#permalink]
 10 Nov 2009, 15:50
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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
Last edited by Bunuel on 31 Jul 2012, 13:51, edited 2 times in total.
Edited the question and added the OA
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Re: Permutation Problem [#permalink]
10 Nov 2009, 15:57
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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 Answer e. 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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Re: Permutation Problem [#permalink]
10 Nov 2009, 15:59
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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 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.
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Re: Stock exchange [#permalink]
01 Nov 2010, 19:19
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, 09:48
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, 09:18
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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, 10:25
26 + 26^2 + 26^3 = 26+676+17576=18278
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Re: Permutation Problem [#permalink]
15 Feb 2013, 03:51
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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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!
DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!
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Re: A certain stock exchange designates each stock with a 1, 2 [#permalink]
15 Feb 2013, 04:04
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, 11:12
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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A certain stock exchange designates each stock with a 1, 2, or 3
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23 Mar 2013, 11:12
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