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  4. 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

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Q. 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

Permutations and Combinations

Solution:

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 are $(1,8),(2,$, 7), $(3,6),(4,5)$. With each pair left number of $7$ -digit number is $7 !$ So with all $4$ pairs, total number is $4 \times 7 !$.