At a party, there were 105 handshakes. If each person shook hands exactly once with every other person, how many people were at the party?

If there are N folks, then

n-1 + N-2 + n-3 +....1 handshakes
take 4 people:
3+2+1= 6 handshakes.

10 people is
9+8+7+6+5+4+3+2+1= 45

14= 13+12+11+ 10 + number above
= 13+12+11+10+45= 91
you want 105, so try
15= 14+ 91

So the answer is 15? But still I don't get how you solve it

To find the number of people at the party, we need to determine the value of "n" in the equation for handshakes: n(n-1)/2. Here, "n" represents the number of people at the party.

Given that there were 105 handshakes, we can set up the equation as follows:

n(n-1)/2 = 105

To solve this equation, we can simplify it by multiplying both sides by 2:

n(n-1) = 210

Expanding the equation gives us:

n^2 - n = 210

Rearranging the equation to solve for n gives:

n^2 - n - 210 = 0

This is a quadratic equation, which can be solved by factoring or using the quadratic formula. In this case, let's factor the equation:

(n - 15)(n + 14) = 0

Setting each factor equal to zero and solving for n gives:

n - 15 = 0 --> n = 15
or
n + 14 = 0 --> n = -14

Since the number of people cannot be negative, we discard n = -14.

Therefore, the number of people at the party is n = 15.

So, there were 15 people at the party.