Question

How many integers with at least one eight and at least one nine as digits are there between $1$ and $10000$.

• A. $800$
• B. $974$
• C. $900$
• D. $875$

Answer 1

The correct option is B $974$

Numbers containing both $8$ and $9$ $=$ Total $-$ (Number not containing $8$ and $9$ )

$\Rightarrow$ Total numbers $=10000$

$\Rightarrow$ Numbers not containing $8:$ We can fill the $4$ places with any number other than $8=9\times 9\times 9\times 9=9^4$

$\Rightarrow$ Similarly for numbers not containing $9=9^4$

$\Rightarrow$ Numbers not containing both $8$ and $9$ $=8^4$

$\Rightarrow$ Required $={2.9}^4-8^4$

$\Rightarrow$ Total Req. $={10}^5-{2.9}^4+8^4$

$\therefore$ Numbers containing $8$ and $9$ $=974.$

Hence, the answer is $974.$