Question

seven digits from the digits 1,2,3,4,5,6,7,8,9 are written in random order. the probability that this seven digit no. is divisible by 9 is ?

Answer 1

Recall the fact that an integer is divisible by $9$ if the sum of its digits is divisible by $9$.

Since the sum of first $9$ natural numbers is $\frac{9(9+1)}{2}=45$, which is divisible by $9$, it must be the case that the sum of the two integers that we don't pick to form the seven digit number is divisible by $9$.

Number of ways of choosing two integers from $9$ integers :  $^9C_2=36$

Number of two digit pairs whose sum is divisible by $9$ : $\{(1,8),(2,7),(3,6),(4,5)\}=4$

Simply take the ratio to get the probability that the seven digit number so formed is divisible by $9$ : 
$\dfrac{4}{36}=\dfrac{1}{9}$

Answer 2

Answer:

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