Question: Two members of a club are to be selected to represent the club at a national meeting. If there are 190 different possible selections of the 2 members, how many members does the club have?

1. 20
2. 27
3. 40
4. 57
5. 95

Correct Answer: (A)

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:
Given:

• 2 members are selected to represent the club
• There are 190 different possible selections of the 2 members

Find out:

• The number of members the club has.

As per the condition of the question, we can write,
nC2=190
=> n!/ {2!∗(n−2)!} =190.
Now, we know that n!=(n−2)!∗(n−1)∗n
Therefore, {(n−2)!∗(n−1)∗n} / {2!∗(n−2)!} = 190
=> (n−1)∗n/2 = 190
=> (n−1)n= 380
=> n=20.
Hence, the number of members the club has = 20.

Approach Solution 2:

The problem statement discloses that:
Given:

• 2 members are selected to represent the club
• There are 190 different possible selections of the 2 members

Find out:

• The number of members the club has.

We can say, nC2, or n(n-1)/2 = 190
Therefore, n(n-1) = 380
Therefore, the prime factorization of 380 = 2*2*5*19
Thus it is obvious that 19 can be considered as one of the numbers (n-1), and then the other is n which is equal to 20.

Approach Solution 3:

The problem statement reveals that:
Given:

• 2 members are selected to represent the club
• There are 190 different possible selections of the 2 members

Find out:

• The number of members the club has.

The order of choosing the two members is of no significance; therefore, we can use combinations. Let n = the total members of the club, then we can write,
nC2 = 190
nC2 = n!/[(n - 2)! x 2!] = [ n x (n-1) x (n-2) x (n-1) x … x 1] / {[(n-2) x (n-1) x … x 1] x 2!}
We analyse that each of the factors in the numerator cancels with those in the denominator, excluding n x (n-1). Therefore, we get:
(n)(n-1)/2 = 190
=>n^2 - n = 380
=>n^2 - n - 380 = 0
=>(n - 20) (n + 19) = 0
Therefore, n = 20 or n = -19
Since n cannot be a negative integer, then n must be 20.
Hence, the number of members the club has = 20.

“Two members of a club are to be selected to represent the club''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Advanced Questions”. GMAT Problem Solving questions analyze the candidates’ skills in solving mathematical problems. GMAT Quant practice papers cite varieties of questions that will enable the candidates to improve their mathematical skills.

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