taylor was having a party.she invited 20 friends to her party, if each person shook hands with every other person .how many handshakes would there be? please explain,thanks
21 * 20 = ?
Each of the 20 people will shake hands with 19 other people
20(19)= 380
but that would include duplicates such as A shaking hands with P, and P shaking hands with A, which is the same handshake
so number of handshakes = 20(19)/2 = 380/2 = 190
or
we are choosing pairs from 20
= C(20,2) = 190
Thanks, Reiny -- I goofed and didn't consider that we can't count A and P and P and A.
I included the hostess -- so we have 21 people.
Thanks Ms Sue, and I didn't catch the hostess part, only saw the 20.
So let's go with your answer ÷ 2 = 21(20)/2 = 210
To find out how many handshakes there would be at Taylor's party, we can use a combination formula.
In this scenario, Taylor invited 20 friends. Now, let's think about the process of handshaking. When one person shakes hands with another person, it is essentially a pairing of two individuals.
To determine how many ways we can pair up the 20 friends, we need to calculate the number of combinations of 20 people taken 2 at a time (since handshaking requires two participants at once). The formula for combinations is:
C(n, r) = n! / (r!(n - r)!)
Where:
n = number of total items/people
r = number of items/people taken at a time
In this case, n = 20 and r = 2, as we are choosing 2 friends at a time to shake hands. So, substituting these values into the formula:
C(20, 2) = 20! / (2!(20 - 2)!)
Simplifying the expression further:
C(20, 2) = 20! / (2! * 18!)
Since 2! (2 factorial) is 2 * 1 = 2, and 18! (18 factorial) can be canceled out in the numerator and denominator:
C(20, 2) = (20 * 19) / 2
Multipling and simplifying:
C(20, 2) = 380 / 2
C(20, 2) = 190
Therefore, there would be a total of 190 handshakes at Taylor's party.