Question

How many $8$ digit mobile numbers can be formed if at least one digit is repeated?

• A. ${10}^8$
• B. ${10}^7$
• C. ${10}^8{-}^{10}{P}_7$
• D. ${10}^8{-}^{10}{P}_8$

Answer 1

The correct option is D ${10}^8{-}^{10}{P}_8$

At least one digit is repeated $=$ Total $-$ No. digit is repeated

For total permutations,

Any of the digit can be filled by any of the $10$ numbers.

$\Rightarrow$ Total $=10\times \mathrm{10.......8}$ times $={10}^8$

For no digit to be repeated:

Let first digit be any of the $10$ numbers,

Next digit can be any of the $9$ remaining and so on.....

$=10\times 9\times 8\times 7\times 6\times 5\times 4\times 3$

$={}^{10}{P}_8$

$\Rightarrow$ Required $={10}^8-{}^{10}{P}_8$

Hence, the answer is ${10}^8-{}^{10}{P}_8.$