Let n greaterthanorequalto 7 be an integer. Why does the following equality hold

Question

Let n greaterthanorequalto 7 be an integer. Why does the following equality hold: (n 5) (n - 5 2) = (n 2) (n -2 5) Because both sides count the number of pairs (A, B) of subsets of the n people, such that |A| = 2, |B| = 5, and A B. Because both sides count the number of pairs (A, B) of subsets of the n people, such that |A| = 5, |B| = 2, and A B. Because both sides count the number of ways to choose 2 committees in a group of n people, one committee has 5 members, the other committee has 2 members, and a person can be on both committees. Because both sides count the number of ways to choose 2 committees in a group of n people, one committee has 5 members, the other committee has 2 members, and no person can be on both committees.

Explanation / Answer

here we LHS = nC5n-5C2

Here we choose 2 committees, 5 members in one committee and 2 in other.

RHS = cc2n-2c5

Here also we choose 2 committees, 5 members in one committee and 2 in other.

Here while choosing a committee of 2 members from n members and 5 members from remaing n-2 members there ins no common member in both committee.

So the answer is

d)

Because both sides count the number of ways to choose 2 committees in a group of n

People, one comitte has 5 members, the other committee has 2 members, and no person

can be on both committees.

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