if 15 people met at a party and all shook hands with one another, how many hand shakes would there be? (assume no one shakes the same hand twice?

the first person shakes 14 hands.

The second person also shakes 14 hands, but he's already shaken hand #1, leaving only 13 other hands.

Continuing, we see that the total unique hands shaken is 14+13+12+...+1 = 14*15/2 = 105

To determine the number of handshakes, we can use the formula for the number of combinations of 2 items from a set of n items.

In this case, we have 15 people, and we want to know how many ways we can choose 2 people to shake hands.

The formula for combinations is

C(n, r) = n! / (r!(n-r)!)

where n! represents factorial notation.

So, plugging in our values:

C(15, 2) = 15! / (2!(15-2)!)

Calculating each factorial value:

15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
2! = 2 x 1
(15-2)! = 13! = 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Substituting in these values:

C(15, 2) = (15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / [(2 x 1)(13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)]

Calculating:

C(15, 2) = 105

So, there would be a total of 105 handshakes at the party.

To calculate the number of handshakes at the party, we can use a simple formula.

Let's say there are 'n' people at the party. If each person shakes hands with every other person once, then we need to find the sum of the first 'n-1' positive integers since the first person has already shaken hands with everyone else.

The formula for the sum of the first 'n-1' positive integers is:

Sum = (n-1) * (n-1 + 1) / 2

In this case, 'n' represents the number of people at the party, which is 15.

Now, let's substitute 'n' into the formula and solve for the number of handshakes:

Sum = (15-1) * (15-1 + 1) / 2
= 14 * (14 + 1) / 2
= 14 * 15 / 2
= 210 / 2
= 105

Therefore, there would be a total of 105 handshakes at the party.