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A researcher plans to identify each participant in a certain [#permalink]

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17 Jun 2012, 03:13

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A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4 B. 5 C. 6 D. 7 E. 8

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

John has 12 clients and he wants to use color coding to identify each client. If either a single color or a pair of two different colors can represent a client code, what is the minimum number of colors needed for the coding? Assume that changing the color order within a pair does not produce different codes. A. 24 B. 12 C. 7 D. 6 E. 5

The concept is not that hard. We can use combination or trial and error approach.

Combination approach: Let # of colors needed be \(n\), then it must be true that \(n+C^2_n\geq{12}\) (\(C^2_n\) - # of ways to choose the pair of different colors from \(n\) colors when order doesn't matter) --> \(n+\frac{n(n-1)}{2}\geq{12}\) --> \(2n+n(n-1)\geq{24}\) --> \(n(n+1)\geq{24}\) --> as \(n\) is an integer (it represents # of colors) \(n\geq{5}\) --> \(n_{min}=5\).

Trial and error approach: If the minimum number of colors needed is 4 then there are 4 single color codes possible PLUS \(C^2_4=6\) two-color codes --> 4+6=10<12 --> not enough for 12 codes;

If the minimum number of colors needed is 5 then there are 5 single color codes possible PLUS \(C^2_5=10\) two-color codes --> 5+10=15>12 --> more than enough for 12 codes.

Actually as the least answer choice is 5 then if you tried it first you'd get the correct answer right away.

won't this \(C^2_n\) just give you all the pairs available? we need them also ordered....

Notice that we are told that letters in the code should be written in alphabetical order. Now, 2Cn gives different pairs of 2 letters possible out of n letters, but since codes should be written in one particular order (alphabetical), then for each pair there will be only one ordering possible, thus the number of codes out of n letters equals to number of pairs out of n letters.

Re: A researcher plans to identify each participant in a certain [#permalink]

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23 Dec 2012, 12:22

Bunuel wrote:

sarb wrote:

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4 B. 5 C. 6 D. 7 E. 8

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4 B. 5 C. 6 D. 7 E. 8

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

Re: A researcher plans to identify each participant in a certain [#permalink]

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24 Dec 2012, 00:28

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Bunuel wrote:

Practice: try to use the same concept.

Okay here goes,

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be (in alphabetical order); \(nC2\) The # of Triples of distinct letters codes possible would be (in alphabetical order); \(nC3\)

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be (in alphabetical order); \(nC2\) The # of Triples of distinct letters codes possible would be (in alphabetical order); \(nC3\)

Thus

\(nC3 + nC2 + n\)> \(12\)

\(n*(n-1)/2 + n*(n-1)*(n-2)/3*2 + n\)> \(12\)

Simplifying

\(n*(n^2 +5)\)> \(72\)

Only sufficient value of \(n = 4\)

Is it correct?

Correct.

Three letters A, B, and C, are enough for 7<12 codes: A; B; C; AB; AC; BC; ABC.

Four letters A, B, C, and D are enough for 15>12 codes: A; B; C; D; AB; AC; AD; BC; BD; CD; ABC; ABD; ACD; BCD; ABCD.
_________________

Re: A researcher plans to identify each participant in a certain [#permalink]

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22 Nov 2013, 14:15

Bunuel wrote:

sarb wrote:

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4 B. 5 C. 6 D. 7 E. 8

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

I still have a little confuse in your formula \(C^2_n\). I am thinking this should be \(A^2_n\) because the 2-letter code must be in alphabetical order.

Hope to hear from you soon.

Thanks

Last edited by yenpham9 on 22 Nov 2013, 14:22, edited 1 time in total.

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4 B. 5 C. 6 D. 7 E. 8

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

I still have a little confuse in your formula \(C^2_n\). I am thinking this should be \(A^2_n\) because the 2-letter code must be in alphabetical order.

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

I still have a little confuse in your formula \(C^2_n\). I am thinking this should be \(A^2_n\) because the 2-letter code must be in alphabetical order.

Re: A researcher plans to identify each participant in a certain [#permalink]

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22 Nov 2013, 14:37

Bunuel wrote:

ronr34 wrote:

Hi Bunnel

won't this \(C^2_n\) just give you all the pairs available? we need them also ordered....

Notice that we are told that letters in the code should be written in alphabetical order. Now, 2Cn gives different pairs of 2 letters possible out of n letters, but since codes should be written in one particular order (alphabetical), then for each pair there will be only one ordering possible, thus the number of codes out of n letters equals to number of pairs out of n letters.

Hope it's clear.

Hi Bunuel,

From n letters we choose the number of pairs, the result will be \(C^2_n\) which may include 2 kinds of pairs (AB) and (BA). Still confused .

won't this \(C^2_n\) just give you all the pairs available? we need them also ordered....

Notice that we are told that letters in the code should be written in alphabetical order. Now, 2Cn gives different pairs of 2 letters possible out of n letters, but since codes should be written in one particular order (alphabetical), then for each pair there will be only one ordering possible, thus the number of codes out of n letters equals to number of pairs out of n letters.

Hope it's clear.

Hi Bunuel,

From n letters we choose the number of pairs, the result will be \(C^2_n\) which may include 2 kinds of pairs (AB) and (BA). Still confused .

Maybe the following example would help. Consider 4 letters {a, b, c, d}. How many 2-letter words in alphabetical order are possible? The answer is \(C^2_4=6\): ab; ac; ad; bc; bd; cd.
_________________

Re: A researcher plans to identify each participant in a certain [#permalink]

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22 Nov 2013, 14:53

Hi Bunuel,

From n letters we choose the number of pairs, the result will be \(C^2_n\) which may include 2 kinds of pairs (AB) and (BA). Still confused .[/quote]

Maybe the following example would help. Consider 4 letters {a, b, c, d}. How many 2-letter words in alphabetical order are possible? The answer is \(C^2_4=6\): ab; ac; ad; bc; bd; cd.[/quote][/quote]

Thanks a lot Bunuel. I got it now . Have a nice weekend!

Re: A researcher plans to identify each participant in a certain [#permalink]

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25 Nov 2013, 03:38

Bunuel wrote:

sarb wrote:

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4 B. 5 C. 6 D. 7 E. 8

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself; The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

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