Question

If repetition of digits is allowed, how many five-digit numbers less than 13000 are possible such that the product of the digits of the number is 96?

• A. 24
• B. 30
• C. 36
• D. 42
• E. 46

Answer 1

Answer: (C)

36

Answer 2

$96 = 2^5 \times 3$

Possible $5$-digit numbers less than $13000$:

$11268, 11348, 11446, 12238, 12246, 12344$

For $11268$, if the number is of the form $11xxx$, we can arrange $2, 6$ and $8$ in $3! = 6$ ways

If the number is of the form $12xxx$, we can arrange $1, 6$ and $8$ in $3! = 6$ ways

So, $11268$ can be arranged in $3! + 3! = 6 + 6 = 12$ ways.

For $11348$, the number has to be of the form $11xxx$

So, $3, 4$ and $8$ can be arranged in $3! = 6$ ways

For $11446$, the number has to be of the form $11xxx$

So, $4, 4$ and $6$ can be arranged in = $\frac{3!}{2!} = 3$ ways

For $12238$, the number has to be of the form $12xxx$

So, $2, 3$ and $8$ can be arranged in $3!$ ways = $6$ ways

Similarly, $12246$ can be arranged in $3!$ ways = $6$ ways

And $12344$ can be arranged in = $\frac{3!}{2!} = 3$ ways

Thus, total number of $5$-digit numbers = $12 + 6 + 3 + 6 + 6 + 3 = 36$

Hence, option C is the correct answer.