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

(1) So from this statement, we can get that:

if there are say 2 symbols, then all the possible code words will be formed by using either one symbol or two symbols (without repetition) taking all possible arrangements.
If there are say 3 symbols, then all the possible code words will be formed by using either one symbol or two symbols or three symbols, again without repetition, taking all possible arrangements. But this just explains how to form code words using symbols, doesnt tell us anything about number or symbols or number of words.

Not Sufficient.

(2) We are given number of code words, but no logic about how symbols are used to form code words. Not Sufficient.

Combining the statements, we can do some trial/error. If say there are two symbols A & B, then taking one symbol at a time, we can form two words only:A and B (basically 2P1). Taking 2 symbols at a time, we can form AB or BA, so two more words (basically 2P2). Total 4 words, not 15.

Now lets take 3 symbols at a time, say A, B, C. Taking one symbol at a time, there are three words: A, B, C. (3P1)
Taking two symbols at a time, there are six words: AB, BA, AC, CA, BC, CB. (3P2)
Taking three symbols at a time, there are again six words: ABC, ACB, BAC, BCA, CAB, CBA. (3P3)
So total words = 3+6+6 = 15.

If we take more symbols, number of code words will definitely be more than 15. So the only possibility that leads of exactly 15 code words is with 3 symbols only.
Sufficient to answer the question with YES.

Hence C answer.

i have a feeling the question is missing some word ?

dave13
You have got very good Sentence correction skills
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(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.

Answer (C)

I think we should use permutation. 3P1 + 3P2 + 3P3 = 15 for set containing 3 code words.

VeritasKarishma chetan2u

How would you approach this question? I could not understand what exactly is the ask here...
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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.
Answer (C)
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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

IMO C

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