A ceirtain stock exchange designates each item with a 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 letter used in a different order constitue a different code, how many different stocks is it possible to uniquely designate with these codes ?
2951
8125
15600
16302
18278
Combinations , permutations ?
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Single Letter Code = 26sapuna wrote:A ceirtain stock exchange designates each item with a 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 letter used in a different order constitue a different code, how many different stocks is it possible to uniquely designate with these codes ?
2951
8125
15600
16302
18278
Double Letter Code = 26 x 26 = 676
Three Letter Code = 26 x 26 x 26 = 17576
Total Such Codes = 17576 + 676 + 26 = 18278
Answer: Option E
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- sapuna
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I thought as much. First, I thought that the correct score should end in what 6 multiplied by 6 is..which is 6 but there is no such answer. Then I figured I we have to add them and not multiply.
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My approach is similar to that of Bhoopendra, with a TWIST at the end.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) = 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
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A 1-digit code can be created in 26 ways, a 2-digit code in 26^2 ways, and a 3-digit code in 26^3 ways.sapuna wrote:A ceirtain stock exchange designates each item with a 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 letter used in a different order constitue a different code, how many different stocks is it possible to uniquely designate with these codes ?
2951
8125
15600
16302
18278
Thus, the number of ways to create the 3 codes is:
26 + 26^2 + 26^3
We should recognize that 26, 26^2, and 26^3 all have units digits of 6. Thus, the sum of those 3 numbers will have a units digit of 8. The only answer choice that has a units digit of 8 is choice E. Thus, the answer must be 18,278.
Answer: E
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