Question

The sum of the digits of a seven digit number is $59$.Find the probability that this number is divisible by $11$.

• A. $\frac{5}{21}$
• B. $\frac{4}{21}$
• C. $\frac{3}{22}$
• D. $\frac{3}{29}$

Answer 1

The correct option is B $\frac{4}{21}$

Given that the sum of digits of seven digit number is $59$

The possible combinations are $9999995,9999986,9999977,9999887,9998888$

Therefore total number of seven digit numbers whose sum is $59$ are $\frac{7!}{6!}+\frac{7!}{5!}+\frac{7!}{5!2!}+\frac{7!}{4!2!}+\frac{7!}{4!3!}=7+42+21+105+35=210$

The condition for the number to be divisible by $11$ is Sum of odd terms minus sum of even terms must be divisible by $11$

The number of possibilities that the number is divisible by $11$ is $0+6+6+24+4=40$ $($ the number $9999995$ has $0$ possibilities , the number $9999986$ has $6$ possibilities and so on$)$

So the probability is $\frac{40}{210}=\frac{4}{21}$