What is the number of different ways to choose a chairman, : GMAT Problem Solving (PS)
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What is the number of different ways to choose a chairman,

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What is the number of different ways to choose a chairman, [#permalink]

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New post 28 May 2010, 13:09
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A
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What is the number of different ways to choose a chairman, two deputies, and two assistants for the class committee out of 7 students up for elections.

Which solution is correct ? And Why ?

A) \(7C1\) * \(6C1\) * \(5C1\) * \(4C1\)* \(3C1\)
B) \(7C1\) * \(6C2\) * \(4C2\)
[Reveal] Spoiler: OA
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Re: Choose committee [#permalink]

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New post 28 May 2010, 14:09
nverma wrote:
What is the number of different ways to choose a chairman, two deputies, and two assistants for the class committee out of 7 students up for elections.

Which solution is correct ? And Why ?

A) \(7C1\) * \(6C1\) * \(5C1\) * \(4C1\)* \(3C1\)
B) \(7C1\) * \(6C2\) * \(4C2\)


Correct answer is B: \(C^1_7*C^2_6*C^2_4=630\).

\(C^1_7\) - # of ways to choose 1 chairmen out of 7;
\(C^2_6\) - # of ways to choose 2 deputies out of 6 members left;
\(C^2_4\) - # of ways to choose 2 assistants out of 4 members left.

Answer A is not correct because it's counting # of different ways to choose 2 deputies out of 6 as \(C^1_6*C^1_5\) (and next # of ways to choose 2 assistants out of 4 as \(C^1_4*C^1_3\)) which is not right. \(C^1_6*C^1_5\) will have duplications in it and needs to be divided by 2! (# of peoples), which then gives the same answer as \(C^2_6\).

Consider this: in how many different ways we can choose 2 different letters out of A, B, and C?

AB
AC
BC

Only 3, which can be obtained by \(C^2_3=3\), another way would give incorrect answer - \(C^1_3*C^1_2=6\).

The original question can be solved in another way:
Members: 1 - 2 - 3 - 4 - 5 - 6 - 7. Positions: C (chairmen), D (deputy), D (deputy), A (assistant), A (assistant), N (no position), N (no position): CDDAANN. # of ways to assign each letter (each position) to the members would be the # of permutations of 7 letters CDDAANN = \(\frac{7!}{2!2!2!}=630\).

1-2-3-4-5-6-7
C-D-D-A-A-N-N
D-C-D-A-A-N-N
D-D-C-A-A-N-N
...
...

Hope it helps.
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Re: Choose committee   [#permalink] 28 May 2010, 14:09
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