Question

Let $N$ be the number of six digit numbers that can be formed by using the digits $1,2,3,4$ such that all the digits should appear but exactly two digits can repeat. Then the number of non-trivial divisors of $N$ is

• A. $31$
• B. $32$
• C. $30$
• D. $28$

Answer 1

The correct option is C $30$

Given digits $:1,2,3,4$
$6$ digit numbers to be formed, exactly $2$ digits can repeat
i.e., in the form $1,2,3,3,4,4,\dots$
$2$ repeated digits can be selected in ${}^4{C}_2$ ways.

Number of required numbers,
$N={}^4{C}_2\times \frac{6!}{2!2!}$
$\Rightarrow N=1080=2^3\cdot 3^3\cdot 5^1$
$\therefore$ Number of divisors of $N$ is $(3+1)(3+1)(1+1)=32$
Hence, number of non-trivial divisors of $N$ is $30$ (except $1$ and itself)