For an international Mathematics Olympiad, six delegates

Question

For an international Mathematics Olympiad, country D will send six delegates in total — two will be supervisors and four will be contestants. There are 210 ways in which the six delegates can be chosen and there are seven candidates competing for the four contestants’ places available. How many candidates are competing for the two supervisors’ slots available?

A. 1
B. 2
C. 3
D. 4
E. 5

A. 1
B. 2
C. 3
D. 4
E. 5

bhatiamanu05 · Answer

We can solve this question by using options:

Option A wrong as 2 supervisor need to be selected. 
Going by statement: 4 contestent choosen from 7 -- 7C4 == 35
and total ways =210
Consider 2 supervisor will be choosen from : X people.
So as per question: 35*X= 210
X= 6
So by using options, 
Option A already decided wrong.
Option B 2C2 will be 1 not 6.
Option C 3C2 will be 3 not 6
 Option D 4C2 will be 6 correct answer 
Option E 5C2 will be 10 not 6.

Bunuel · Answer

2 supervisors out of x;
4 contestants out of 7.

C^2_x*C^4_7=210  -->  \frac{x!}{2!(x-2)!}*35=210  --> \frac{x(x-1)}{2}*35=210  -->  x(x-1)=12  -->  x=4  (discard the negative root).

Answer: D.

ScottTargetTestPrep · Answer

We can let n = the number of candidates vying for the 2 supervisors’ slots and create the equation:

nC2 x 7C4 = 210

=  / {  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 nC2 = n(n - 1)/2]

n(n - 1)/2 x (7 x 6 x 5 x 4)/(4 x 3 x 2) = 210

(n^2 - n)/2 x 35 = 210

(n^2 - n)/2 = 6

n^2 - n = 12

n^2 - n - 12 = 0

(n - 4)(n + 3) = 0

n = 4 or n = -3

Since n can’t be negative, then n must be 4.

Answer: D

AndyRugger · Answer

Solving for the answer algebraically and discarding the negative root is a pretty neat trick. That's definitely more elegant than my way.

However, once you've calculated 7C4 as 35 and you realize that 35 * X = 210 and that X must equal 6 = ?C2, it might be more intuitive to back solve.

Choices A, B, C are immediately out because there's no way to get 6 out of a combination formula with those inputs. So you test the next answer D = 4, and wallah, it's the correct one!

For an international Mathematics Olympiad, six delegates

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