JEE Main 2014 — Permutation Combination Question with Solution

We know that any number is divisible by 9 if sum of the digits of the number is divisible by 9 . Now sum of the digits from 0 to 9 $$\begin{array}{rl}& =0+1+2+3+4+5+6+7+8+9\\ & =45\end{array}$$ Hence to form 8 digits numbers which are divisible by 9 , a pair of digits either 0 and 9,1 and 8,2 and 7,3 and 6 or 4 and 5 are not used.
Hence total number of 8 digits numbers which are divisible by 9 $$\begin{array}{rl}& =8\times (7!)+7\times (7!)+7\times (7!)+7\times (7!)\\ & +7\times (7!)\\ & =36\times (7!)\end{array}$$