combination is a way of selecting several things out of a larger group, where order does not matter. in our case it is the selection of 2 persons out of a set of 5 people

we need to find all the 2-combinations of 5 element

C(5, 2) = 5!/(2!*3!) = 10

indeed if 1,2,3,4,5 are the people and we denote (1,2) as the handshake between 1 and 2 all the handshakes are written

(1,2)(1,3)(1,4)(1,5)

(2,3)(2,4)(2,5)

(3,4)(3,5)

(4,5)

there are 10 different handshakes in total.