Question

A seven-digit number without repetition and divisible by $9$ is to be formed by using seven digits out of $1,2,3,4,5,6,7,8,9$. The number of ways in which this can be done is

• A. $9!$
• B. $2(7!)$
• C. $4(7!)$
• D. None of these

Answer 1

The correct option is C $4(7!)$

Sum of $7$ digits is a multiple of $9$.
Sum of numbers $1,2,3,4,5,6,7,8,9$ is $45$;

So two left digits should also have sum of $9.$
The pairs of left numbers $(1,8),(2,7),(3,6),(4,5).$
With each pair left number of $7-$digit number is $7!$
So with all $4$ pairs, total number is $4\times 7!$.