Question

The number of different seven digit numbers that can be written using only three digits $1,2$ & $3$ under the condition that the digit $2$ occurs exactly twice in each number is

• A. $672$
• B. $640$
• C. $512$
• D. $noneofthese$

Answer 1

Answer: (A)

$672$

Total number of digits $=3^7$

The $7-$ digit number has exactly two digits $2$

Let's say the number

$\overline{22cdefg},\overline{2b2defg},\overline{2bc2efg}$,..

Number of cases we can put exactly two digits (out of $7$ digits) are $2$ is

$6+5+4+3+2+1=21$cases.

For each case, the remaining $5$ digits can be either $1$ or $3$, so the number of ways for each case is

$2^5=32$ ways

$\therefore$ the number of favorable numbers is

$=21\times 32=672$ numbers