Question

How many numbers lying between $100$ and $10,000$ can be formed with the digits $1,2,3,4,5$, if atleast one digit in the same numbers are repeated?

• A. $750-{}^5{P}_3-{}^5{P}_4$
• B. $750-{}^5{P}_3+{}^5{P}_4$
• C. $750+{}^5{P}_3-{}^5{P}_4$
• D. $750+{}^5{P}_3+{}^5{P}_4$

Answer 1

The correct option is A $750-{}^5{P}_3-{}^5{P}_4$

Total no. of 3 digit numbers formed$=5\times 5\times 5=125$

Total no. of 4 digit numbers $=5\times 5\times 5\times 5=625$

$\therefore$ Total numbers between 100 & 1000 formed $=750$.

Total number of 3 digit numbers without repetition${=}^5{P}_3$

No. of 4 digit numbers without repetition${=}^5{P}_4$

$\therefore$ No. of numbers between 100 & 1000 with at least one repeating digit$=750{-}^5{P}_3{-}^5{P}_4$