﻿ An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without replacement. The number of ways I which this can be done is   : Kaysons Education

# An Eight Digit Number Divisible By 9 Is To Be Formed By Using 8 Digits Out Of The Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Without Replacement. The Number Of Ways I Which This Can Be Done Is

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## Question

### Solution

Correct option is

(36)(7!)

We have 0 + 1 + 2 + 3 … + 8 + 9 = 45.

To obtain an eight digit number exactly divisible by 9, we must not use either (0, 9) or (1, 8) or (2, 7) or (3, 6) or (4, 5). [Sum of the remaining eight digits is 36 which is exactly divisible by 9.]

When, we do not use (0, 9), then the number of required 8 digit numbers is 8!.

When one of (1, 8) or (2, 7) or (3, 6) or (4, 5) is not used, the remaining digit can be arranged in 8! – 7! Ways as  0 cannot be at extreme left.

Hence, there are 8! + 4(8! – 7!) = (36)(7!) numbers in the desired category.

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