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# Are exactly 3 distinct symbols used to create the code words in langua

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Re: Are exactly 3 distinct symbols used to create the code words in langua [#permalink]
exc4libur wrote:
gmatbusters wrote:
Are there exactly 3 distinct symbols used to create the code words in language Q?

(1) The set of all code words in language Q is the set of all possible distinct horizontal arrangements of one or more symbols, with no repetition.
(2) There are exactly 15 code words in language Q

(1) The set of all code words in language Q is the set of all possible distinct horizontal arrangements of one or more symbols, with no repetition: here there could be a number of symbols, ie. 1 symbol {A}=1 arrangement/code word; 2 symbols {A,B}={A,B,AB,BA}=4 arrangements/code words; insufic.

(2) There are exactly 15 code words in language Q: we don't know how many symbols there are, insufic.

(1&2) our set has 15 code words, and they are equal to all possible distinct horizontal arrangements of one or more symbols;
if we have 1 symbol, call this {A}, then our set of "code words" and of "all possible distinct horizontal arrangements" is {A}=1
if we have 2 symbols, call them {A,B}, then our sets will have {A,B,AB,BA}=4
if we have 3 symbols, call them {A,B,C}, then our sets will have {3C1+3C2+3C3=3+3+1}=7
if we have 4 symbols, call them {A,B,C,D}, then our sets will have {4C1+4C2+4C3+4C4=4+6+4+1}=15, this is it.

I think we should use permutation. 3P1 + 3P2 + 3P3 = 15 for set containing 3 code words.
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Are exactly 3 distinct symbols used to create the code words in langua [#permalink]
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GMATBusters wrote:
Are there exactly 3 distinct symbols used to create the code words in language Q?

(1) The set of all code words in language Q is the set of all possible distinct horizontal arrangements of one or more symbols, with no repetition.

(2) There are exactly 15 code words in language Q

Does the language Q have exactly 3 distinct symbols? (for example, English language has exactly 26 distinct letters)

Stmnt 1 tells us that code words are made by horizontal placement of 1 or more of these symbols. And stmnt 2 tells us that there are exactly 15 code words.

If we have 3 distinct symbols in a language (say @, #, \$), how many code words can we make?
We can make a code word by selecting exactly 1 of these 3 symbols. So 3
We can make a code word by selecting exactly 2 of these symbols and arranging them in 2! ways. So 3C2 * 2!
We can make a code word by selecting all 3 of these symbols and arranging them in 3! ways. So 3!
Total = 3 + 3C2*2! + 3! = 15

So yes, language Q has exactly 3 distinct symbols.
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Re: Are exactly 3 distinct symbols used to create the code words in langua [#permalink]
Asked: Are there exactly 3 distinct symbols used to create the code words in language Q?

(1) The set of all code words in language Q is the set of all possible distinct horizontal arrangements of one or more symbols, with no repetition.
No numbers are given
NOT SUFFICIENT

(2) There are exactly 15 code words in language Q
It is not provided how code words are formed
NOT SUFFICIENT

(1) + (2)
(1) The set of all code words in language Q is the set of all possible distinct horizontal arrangements of one or more symbols, with no repetition.
(2) There are exactly 15 code words in language Q
If there are 3 distinct symbols
Number of 1 symbol code words = 3
Number of 2 symbol code words = 3*2 = 6
Number of 3 symbol code words = 3*2*1 = 6
Total codewords = 3 + 6+ 6 = 15
If the number of distinct symbols differ, then Total codewords will be different than 15.
SUFFICIENT

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Re: Are exactly 3 distinct symbols used to create the code words in langua [#permalink]
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Re: Are exactly 3 distinct symbols used to create the code words in langua [#permalink]
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