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.