# Seven digits from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written down non-repeatedly in a random order to form a seven digit number. The probability that this seven digit number is divisible by 9 is (1) 2/9 (2) 7/36 (3) 1/9 (4) 7/12

We know that the integer is divisible by 9 if the sum of its digits is divisible by 9.

Sum of the first 9 natural numbers is

$$\begin{array}{l}n(S) = \frac{9\left ( ( + 1) \right )}{2}\end{array}$$

n(S) = 45

45 is divisible by 9.

So it must be the case that the sum of the two integers we don’t pick is divisible by 9 to form the seven-digit number.

Number of ways of choosing two integers from 9 integers is

$$\begin{array}{l}^{9}C_{2} = 36\end{array}$$

There are 4 of two-digit pairs whose sum is divisible by 9

{(1,8), (2,7), (3,6), (4,5)}

Take the ratio to get the probability that the seven-digit number so formed is divisible by 9.

4/36 = 1/9