How many hand shakes?

Question

there are 110 people at a meeting. They each shake hands with everyone else. How many handshakes were there?

Answer 1

give each person a number

person 110, person 109, person 108,...

person 110 shakes 109 hands

person 109 shakes 108 hands (person 109 already shook hands with person 110)

person 108 shakes 107 hands (person 108 already shook hands with person 109 and person 110)

do you see a pattern ?

you now have to add the numbers 109+108+107+106+...+4+3+2+1+0 (the last person does not shake any hands because everybody else shook his hand-that's why I put in the 0)

how do you add all these numbers ?

use the formula n(n+1)/2 where n=109, the greatest number of handshakes

(109)(110)/2=109*55=5995 handshakes

you use this formula to add any number of consecutive integers beginning with the number 1

1+2+3+4+5+6+7+8+9+10=(10)(11)/2=55

add the numbers from 1 to 100

(100)(101)/2=5050

this formula is contributed to a mathematician named Gauss