The number of sets of two different numbers which can be formed by using numbers between 0 and 120 (both including) so that 60 is their average, is

Let x, y be two numbers such that $\frac{\mathrm{x}+\mathrm{y}}{2}=60\Rightarrow x+y=120$

x and y both can't be equal to or greater than 60. [$\because$ 60 can't be used twice] 

Let $0\le x\le 59\text{ and }61\le y\le 120$

$\therefore$The total number of ways in which x can be chosen ${=}^{60}{C}_1=60$.