Eight friends meet at a party. Each one shakes, hand with each of the other once. The number of possible handshakes is

The answer is C (28). This is a plain combination problem. A handshake is between two distinct people and is mutual, so Ram shaking hands with Shyam is the same event as Shyam shaking hands with Ram. We need the number of ways to choose 2 people out of 8, which is C(8,2) = (8 x 7)/(2 x 1) = 28.

• A (64) is wrong. It is 8 squared, which would count ordered pairs including a person shaking hands with himself. Neither is allowed.
• B (56) is wrong. It is 8 x 7, the number of ordered pairs. It counts each handshake twice, once from each side.
• D (20) is wrong. It does not correspond to any valid count for this setup.

The core concept is combinations, where order does not matter. This is the same idea behind counting games, pairings and committees in quantitative and reasoning questions. In Mains it shows up in disciplined reasoning about data and counting.

Trap: multiplying 8 by 7 and stopping at 56 without halving it for the double counting.