Let Q(n) denotes the number of seven digit numbers divisible by 9 which can be formed by using 7 distinct digits out of 1,2,3,4,5,6,7,8,9. Then which of the following is/are not the value of Q(n)

Let Q(n) denotes the number of seven digit numbers divisible by 9 whic

1.	Let Q(n) denotes the number of seven digit numbers divisible by 9 which can be formed by using 7 distinct digits out of 1,2,3,4,5,6,7,8,9. Then which of the following is/are not the value of Q(n)
Answer» Sum of the given `9` digits ` = 1+2+3+4+5+6+7+8+9 = 45` Now, we have to select `7` digits. The number will be divisible by `9` if sum of those `7` digits is divisible by `9`. Minimum sum of two digits that we do not select `= 1+2 = 3` Maximum sum of two digits that we do not select `= 8+9 = 17` So, the sum of the `7` digits will lie between `45-17` and `45-3`, that is sum will be between `28` and `42`. Now, there is only number `36` that is divisible by `9`. So, the seven digits will have sum `36` in order to the number to be divisible by `9`. It means we have to exclude two numbers those have `9` as the sum. So, these two numbers can be any of the pair `(1,8),(2,7),(4,5),(6,3).` Total seven digits can be arranged in `7!` ways and there are `4` pairs of number that can be excluded. `:. Q(n) = 4(7!)` So, `Q(n)` can not take values `(A),(B) and (D).`