Question

The total number of $5$ digit numbers of different digits in which the digits in the middle is the largest is

• A. $\sum _{n=4}^9\left({}^n{P}_4{-}^{n-1}{P}_3\right)$
• B. $\sum _{n=4}^9{}^n{P}_4$
• C. $\sum _{n=4}^9{}^{n-1}{P}_3$
• D. None of these

Answer 1

The correct option is A $\sum _{n=4}^9\left({}^n{P}_4{-}^{n-1}{P}_3\right)$

The smallest number, which can occur in the middle is $4.$

The number of numbers with $4$ in the middle ${}^4{P_4-}^3{P}_3,$

$(\therefore$ The other four places are to be filled by $0,1,2$ and $3,$ and a number can not begin with $0)$

Similarly, the number of numbers with $5$ in the middle ${}^5{P_4-}^4{P}_3,$ etc.

$\therefore$ The required number of numbers

$=\left({}^4{P_4-}^3{P}_3\right)+\left({}^5{P_4-}^4{P}_3\right)+\left({}^6{P_4-}^5{P}_3\right)+...+\left({}^9{P_4-}^8{P}_3\right)$

$=\sum _{n=1}^9\left({}^n{P_4-}^{n-1}{P}_3\right)$