Divisible Probability...
The sum of the digits of a
digit number is
. The probability that this number is divisible by 11 can be expressed as
where
and
are co-prime positive integers.
Find the value of .
The answer is 119.
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1 solution
77 can be reached in the following ways ---
1) Eight 9's and one 5 which are arranged to give 9!/8! = 9 numbers
2) Seven 9's two 7's which are arranged in 9!/7!2! = 36 numbers
3) Seven 9's one 6 and one 8 which can be arranged in 9!/7! = 72 numbers
4) Six 9's two 8's and one 7which can be arranged in 9!/6!2! = 252 numbers
5) Five 9's four 8's which can be arranged in 9!/5!4! = 126 numbers
Total of 495 numbers possible.
For the number to divisible by 11, the sum of digits in the even places should be 33 while sum of digits in odd places should be 44.
1) 9996 in even places, 99998 in odd places. these can be arranged respectively in 4 and 5 ways = total of 20 numbers
2) 9987 in even places, 99998 in odd places, these can be arranged respectively in 12 and 5 ways so total of 60 numbers
3) 9888 in even places and 99998 in odd places, these can be arranged in 4 and 5 ways so total of 20 numbers.
Total of 100 numbers divisible by 11.
Probaility of 100/495 = 20/99
Sum = 119