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Intern  Joined: 12 May 2012
Posts: 17
Location: United States
Concentration: Technology, Human Resources
A researcher plans to identify each participant in a certain medical  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 60% (01:40) correct 40% (01:52) wrong based on 3898 sessions

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

Originally posted by sarb on 17 Jun 2012, 03:13.
Last edited by Bunuel on 04 Feb 2019, 23:52, edited 2 times in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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47
71
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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.
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Math Expert V
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Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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5
63
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

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Hope it helps.
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Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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13
26
Almost identical question:

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.

Hope it helps.
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Intern  Joined: 27 Aug 2012
Posts: 11
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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Bunnel. Thaks for the reply and merging similar topics. Can u please explain how >= 12 ?
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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SreeViji wrote:
Bunnel. Thaks for the reply and merging similar topics. Can u please explain how >= 12 ?

The number of letters should be enough to make at least 12 codes, thus the number of codes must be more than or equal to 12.
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Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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4
Hi Bunnel

won't this $$C^2_n$$
just give you all the pairs available?
we need them also ordered....
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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5
6
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.
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Manager  Joined: 24 Mar 2010
Posts: 58
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.

Bunuel,

What if the question didn't say 'pair'.

If 3 letter combinations were also permitted. How would you express it in Combination formula?
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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3
1
eaakbari wrote:
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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.

Bunuel,

What if the question didn't say 'pair'.

If 3 letter combinations were also permitted. How would you express it in Combination formula?

Practice: try to use the same concept.
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Manager  Joined: 24 Mar 2010
Posts: 58
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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2
2
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); $$nC3$$

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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2
2
eaakbari wrote:
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); $$nC3$$

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;
BC;
BD;
CD;
ABC;
ABD;
ACD;
BCD;
ABCD.
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Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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1
15
Bunuel wrote:
yenpham9 wrote:
Bunuel wrote:
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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.

Hi Bunnel,

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

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Hope this helps.
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Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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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 .
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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yenpham9 wrote:
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 .

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;
bc;
bd;
cd.
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Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.

we can take 1,2 and 3
like
A, B, C
AB, BC
ABC

Why did you ignored possibility of 3 or 4 alphabets taken together, this will give us 4 letters?
Math Expert V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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honchos wrote:
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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.

we can take 1,2 and 3
like
A, B, C
AB, BC
ABC

Why did you ignored possibility of 3 or 4 alphabets taken together, this will give us 4 letters?

Please read the question carefully: a code consists of either a single letter or a pair of distinct letters written in alphabetical order.
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Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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Lets take A B C D
A
B
C
D
AB
AC
BC
BD
CD
ABC
BCA
CBA

It is alphabetical and all letter for a particular codes are different.
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Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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2
honchos wrote:
Lets take A B C D
A
B
C
D
AB
AC
BC
BD
CD
ABC
BCA
CBA

It is alphabetical and all letter for a particular codes are different.

Please read the question carefully. The stem says that a code can consists of 1 or 2 letters ONLY: a code consists of either a single letter or a pair of distinct letters written in alphabetical order.
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Re: A researcher plans to identify each participant in a certain medical  [#permalink]

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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);

We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$.

Hope it's clear.

I have a questions here:
How did we get from $$n(n+1)\geq{24}$$ to $$n_{min}=5$$ Re: A researcher plans to identify each participant in a certain medical   [#permalink] 25 Feb 2014, 07:57

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