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  • An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits <img alt="0,1,2,3,4, 5,6,7,8,9" src="https://entrancecorner.oncodecogs.com/gif.latex?0%2C1%2C2%2C3%2C4%2C%20

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 in which this can be done is

Option: 1

9 !


Option: 2

2(7 !)


Option: 3

4(7 !)


Option: 4

(36) (7!)


Answers (1)

best_answer

We have
0+1+2+3 \ldots+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 digits can be arranged in 8 !- 7 ! ways. ( 0 cannot be at extreme left)

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

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