12 people at a party shake hands once with everyone in the room. How many handshakes took place?Choices:- 77 54 78 66

The solution to this problem can be found using the formula for combinations in mathematics. The formula for combinations is nCr = n! / r!(n-r)!, where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

In this case, we have 12 people and we want to find out how many ways we can choose 2 people to shake hands. So, n = 12 and r = 2.

First, calculate the factorial of n (n!): 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 479,001,600

Next, calculate the factorial of r (r!): 2! = 2 × 1 = 2

Then, calculate the factorial of (n-r): (n-r)! = (12-2)! = 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

Now, substitute these values into the formula and solve: nCr = n! / r!(n-r)! = 479,001,600 / (2 × 3,628,800) = 66

So, 66 handshakes took place at the party. The correct choice is 66.