Question

The total number of $5$ digit numbers of different digits in which the digit 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 ${{}^{4}p}_4-{{}^{3}p}_3$ (as the other $4$ digits are to be filled by $0,1,2,3$ and a number cannot begin with $0$)

Similarly, the number of numbers with $5$ in the middle ${}^5{P}_4{-}^4{P}_3$, and so on.

Thus the required numbers

${}^4{P}_4{-}^3{P}_3{+}^5{P}_4{-}^4{P}_3{+}^6{P}_4{-}^5{P}_3....{+}^9{P}_4{-}^8{P}_3={\sum }_{n=4}{(}^n{P}_4{-}^{n-1}{P}_3)$