Class 11

Math

Algebra

Permutations and Combinations

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

- $10_{8}$
- $10_{7}$
- $10_{8}−_{10}P_{7}$
- $10_{8}−_{10}P_{8}$

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.

$⇒$ Total $=10×10.......8$ 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.....

$=10×9×8×7×6×5×4×3$

$=_{10}P_{8}$

$⇒$ Required $=10_{8}−_{10}P_{8}$

Hence, the answer is $10_{8}−_{10}P_{8}.$