A question on counting I'm having difficulty answering (Qb iii)
Question:
(a) Write an expression for the number of sets S which contain 10 elements, each of which is
an integer between 1 and 20.
Even more explicit hint... the smallest possible value of $t(A)$ would happen when the subset is $\{1,2\}$ since this is the set using the two smallest available elements while the largest possible value of $t(A)$ would happen when the subset is $\{19,20\}$ since this is the set using the two largest available elements.

 The possible outcomes of $t(A)$ are then $3,4,5,\dots,37,38,39$

You correctly noted however that there are $\binom{10}{2}=45$ distinct two-element subsets of $S$ however

 And if we compare the number of possible two-element subsets of $S$ to the number of possible values of $t(A)$ we see that one is larger than the other which means...

