Two members of a club are to be selected to represent the club at a na

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?

A. 20 
B. 27 
C. 40 
D. 57 
E. 95

I know I can just plug in answer choices in to the formula n!/(n-r)!*r!=190

n=20

but I saw a explanation on this problem where they showed this step

n!/(n-2)! = n(n-1)

How is this step possible ?

Bunuel · Accepted Answer

Welcome to GMAT Club. Below is an answer to your question.

C^2_n=190  -->  \frac{n!}{2!*(n-2)!}=190 . Now, notice that  n!=(n-2)!*(n-1)*n , so  \frac{(n-2)!*(n-1)*n}{2!*(n-2)!}=190  -->  \frac{(n-1)*n}{2}=190  -->  (n-1)n=380  -->  n=20 .

Answer: A.

P.S. Please always post answer choices for PS questions. On the PS section one should always look at the answer choices before starts to solve a problem. They might often give a clue on how to approach the question.

energetics · Answer

nC2, or n(n-1)/2 = 190
n(n-1) = 380
We can use prime factorization: 380 = 2*2*5*19
Becomes pretty obvious that 19 is one of the numbers (n-1) and the other is n=20.

nishantmehra01 · Answer

I have done this problem by substituting numbers

We have n!/2!(n-2)! = 190

or n(n-1)/2 = 190

n=20 satisfies the equation

ScottTargetTestPrep · Answer

The order of selecting the two representatives is of no importance; thus, we use combinations. Letting n = the total number of members of the club, we can create the equation:

nC2 = 190

nC2 = n!/  =  / {  x 2!}

We see that all the factors in the numerator cancel with those in the denominator, except n x (n-1). Thus, we have:

(n)(n-1)/2 = 190

n^2 - n = 380

n^2 - n - 380 = 0

(n - 20)(n + 19) = 0

n = 20 or n = -19

Since n can’t be negative, then n must be 20.

Answer: A

IanStewart · Answer

(n)(n-1)/2! = 190
(n)(n-1) = 380
n^2 - n = 380

Since n is a fairly large positive integer, n^2 is much larger than n, so we want a value of n^2 that is very close to 380. Glancing at the answer choices, n = 20 is the only realistic candidate, and it's easy to plug it back in to confirm it works.

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Two members of a club are to be selected to represent the club at a na

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