The number of all three element subsets of the set $\left\{a_{2,},a_2,a_3\mathrm{....}{a}_n\right\}$ contain $a_3,$ is

The number of three element subsets containing $a_3$  is equal to the number of ways of selecting 2 elements out of $n–1$ elements.

So, the required number of subsets $={}^{n–1}C{}_2.$