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AA vines

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· 1 Months ago

permutation n combination (Question: 45209.0 )

I can't understand this question please explain it in more detail

Question:
An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0,1,...9 without repetition. The number of ways in which this can be done is:
Options:
A) 9!
B) 2(7!)
C) 4(7!)
D) 36(7!)
E) 26(2!)
Solution:
Sum of the digits of a number divisible by 9 is also divisible by 9. Now sum of digits 0,1,2,3,...9 is 45 which is divisible by 9.

So, two digits out of 10 digits given will be omitted in such a way that their sum should be also divisible by 9.

So, omitted digits will be (0,9); (1,8); (2,7); (3,6) and (4,5). In the first case there will be 8! numbers divisible by 9 and in the last four case there will be 7(7!) way due to presence of 0.

So, total number of ways = 8! + 4 x 7(7!) = 36(7!)

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