Question

If in a hall there are 18 persons then how many handshakes are possible.

Answer 1

Handshakes = (Number of people)^2

=(18)^2= 324

Answer 2

how many ways can you choose 2 people out of 18 people ?

18!/2!(18-2)!=18!/2!(16)!=(18*17)/2=306/2=153 ways, or 153 handshakes

using arithmetic...

the first person shakes 17 hands

the second person shakes 16 hands (person 1 and person 2 already shook hands)

the third person shakes 15 hands (person three shook hands with persons 1 and 2 already)

the fourth person shakes 14 hands...

the fifth person shakes 13 hands...

then 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

note that the last person doesn't shake hands with anybody because everybody else already shook the last (18th) person's hand

add the handshakes

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17=153 handshakes

don't forget that each person shakes hands with everybody else only one time !!!