A number n ≥ 5 of persons meet at a party, and some of them shake hands. It is known that at least one person does not shake hands with everybody. What is the maximum number of persons that may

have shaken hands with everyone?

# Maximum number of persons

**Nirbhay**#1

10 - 9 hand shakes (the other 9 people)

#9 - 8 hand shakes (he doesn’t need to reshake #10’s hand)

#8 - 7 hand shakes (he doesn’t need to reshake #10’s and #9’s hand)

…

#3 - 2 hand shakes (he doesn’t need to reshake all the other’s hands and #2 & #1 are the only ones he has not

#2 - 1 hand shake (he only had #1 left that he has not shaken hands with)

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