vaibhav101 wrote:how many different 5 person teams can be formed from a group of x individuals?

1) if there had been x+2 individuals in the group, exactly 126 different 5 person teams could have been formed.
2) if there had been x+1 individuals in the group, exactly 56 different 3 person teams could have been formed.

We have to find out the number of different 5-person teams that can be formed from a group of x individuals.

Let's take each statement one by one.

1) if there had been x+2 individuals in the group, exactly 126 different 5 person teams could have been formed.

=> (x + 2) C5 = 126

{(x+2).(x+1).x.(x-1).(x-2)}/{1.2.3.4.5} = 126
(x+2).(x+1).x.(x-1).(x-2) = 1.2.3.4.5.(126)

We see that LHS, (x+2).(x+1).x.(x-1).(x-2) is a product of 5 consecutive positive integers. If we are able to arrange 1.2.3.4.5.(126) such that it is also a product of 5 consecutive positive integers, we would get the unique answer.

Let's try to manipulate 1.2.3.4.5.(126).

1.2.3.4.5.(126) = 2.3.4.5.(2.7.9) = 5.(3.2).7.(4.2).9 = 5.6.7.8.9 = product of 5 consecutive positive integers

=> x = 7. Sufficient.

2) if there had been x+1 individuals in the group, exactly 56 different 3 person teams could have been formed.

=> (x+1)C3 = 56

{(x+1).x.(x-1)}/1.2.3 = 7.8
(x+1).x.(x-1) = 6.7.8

=> x = 7. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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