Question

The total number of five digit numbers consisting of distinct digits in which the digit in the middle is the largest is :

• A. $\sum _{n=4}^9{}^n{P}_4$
  
• B. $\sum _{n=4}^{9}n\left({}^n{P}_3\right)$
• C. $30(3!)$
• D. $\sum _{n=3}^{8}n\left({}^n{P}_3\right)$

Answer 1

The correct option is D $\sum _{n=3}^{8}n\left({}^n{P}_3\right)$

$\begin{array}{cccc}\text{Middle Digit}& \text{Digits available}& & \\ & \text{for remaining places}& \text{Number of ways}& \\ & & \text{filling remaining four places}& \\ 4& 0,1,2,3& 3\times {}^3{P}_3& \\ 5& 0,1,2,3,4& 4\times {}^4{P}_3& \\ 6& 0,1,2,3,4,5& 5\times {}^5{P}_3& \\ 7& 0,1,2,3,4,5,6& 6\times {}^6{P}_3& \\ 8& 0,1,2,3,4,5,6,7& 7\times {}^7{P}_3& \\ 9& 0,1,2,3,4,5,6,7,8& 8\times {}^8{P}_3& \end{array}$
Total number of required ways is $3\times {}^3{P}_3+4\times {}^4{P}_3+5\times {}^5{P}_3+6\times {}^6{P}_3+7\times {}^7{P}_3+8\times {}^8{P}_3$