JEE Main 2019 — Permutation Combination Question with Solution

Each of the beams connects the top of two non-adjacent pillars, hence we have to find in how many ways we can select the required two pillars which are non-adjacent out of the given $20$ pillars.

As, ${}_{ }{}^{20}C_2$ is the number of beams connecting any two pillars whereas $20$ is the number of adjacent beams.

Thus, the total number of beams $={}_{ }{}^{20}C_2-20$

Using ${}^{n}C_r=\frac{n!}{r!\left(n-r\right)!},$

$=\frac{20!}{2!\times 18!}-20$

$=\frac{20\times 19\times 18!}{2!\times 18!}-20$

$=190-20$

$=170.$