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What is the difference between the number of three-member

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What is the difference between the number of three-member

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Princeton Review

What is the difference between the number of three-member committees that can be formed from a group of nine members and the total number of ways there are to arrange the members of such a committee?

A. 0
B. 84
C. 252
D. 420
E. 504

OA D.

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AAPL wrote:
Princeton Review

What is the difference between the number of three-member committees that can be formed from a group of nine members and the total number of ways there are to arrange the members of such a committee?

A. 0
B. 84
C. 252
D. 420
E. 504
The number of three-member committees that can be formed from a group of nine members
Since the order in which we select the 3 people does not matter, we can use combinations.
We can choose 3 people from 9 people in 9C3 ways
9C3 = (9)(8)(7)/(3)(2)(1) = 84 three-member committees

The total number of ways there are to ARRANGE the members of such a committee
We can arrange n objects in n! ways
So, we can arrange the 3 people (in the committee) in 3! ways (= 6 ways)

DIFFERENCE = 84 - 6 = 78
78 is not among the answer choices.

hmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm!

It appears (given the official answer) that we're supposed to arrange the 3 members in each of the 84 three-member committees
So, for each of the 84 three-member committees, we can arrange the three people in 6 ways
So, the total number of arrangements = (84)(6) = 504
So, the DIFFERENCE = 504 - 84 = 420 (D)

IMO, this question is too ambiguous to be GMAT-worthy.

Cheers,
Brent

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Hi All,

To start, I agree with Brent; the wording of this prompt is 'clunky' - and the GMAT writers word their questions in a far more rigorous and specific fashion than what we see here. That having been said, the basic concepts involved here are Combinations and Permutations.

We're asked for the difference between the number of three-member committees that can be formed from a group of nine members and the total number of ways there are to arrange the members of such a committee. The intent of this question is to ask for the difference in the number of possible 3-person groups and the number of ways to arrange 3 of the 9 people 'in a row.'

For the number of 3-person groups, we can use the Combination Formula: N!/K!(N-K)! = 9!/3!(9-3)! = (9)(8)(7)/(3)(2)(1) = 504/6 = 84

The number of ways to arrange 3 of the 9 people in a row = (9)(8)(7) = 504

The difference is 504 - 84 = 420

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

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Quote:
[Possible alternate question stem to avoid Brent´s and Rich´s (and also my) insatisfaction]

What is the difference between the number of ways to select a three-member committee from a group of nine members, taking or not taking into account the order of selection of these members?

A. 0
B. 84
C. 252
D. 420
E. 504
$$? = 3!C\left( {9,3} \right) - C\left( {9,3} \right)\,\,\mathop = \limits^{\left( * \right)} \,\,\,84\left( {3! - 1} \right) = 5 \cdot 84 = 420$$
$$\left( * \right)\,\,\,C\left( {9,3} \right) = {{9 \cdot 8 \cdot 7} \over {3 \cdot 2}} = 3 \cdot 4 \cdot 7 = 84$$
This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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