Nice photo you have there.
We have n persons. Handshake involves selection of two persons. Order of selection of these two is not important. That is, it is the combination of the two persons, makes one handshake.
Hence there are nC₂ hand shakes in a group of n persons.
nC₂ = n! / 2! (n-2)! = n(n-1)/2 = 91
n (n - 1) = 2 * 7 * 13
n (n - 1) = 14 * 13
hence, n = 14