This is a problem of combinations, where the order of shaking hands does not matter (person A shaking hand with person B is the same as person B shaking hand with person A). We need to choose 2 people out of 10 for each handshake. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to choose. In this case, n=10 and r=2. So, 10C2 = 10! / (2!(10-2)!) = 10! / (2!8!) = (10 * 9) / (2 * 1) = 90 / 2 = 45.