The total number of five-digit numbers of different digits in which the digit in the middle is the largest is a. `sum_(n=4)^9 ^n P_4` b. `33(3!)` c. `30(3!)` d. `sum_(n=3)^8 n^n P_3`

Options

A. `2^(2)*3^(2)*7^(2)`
B. `2^(3)*3*7^(3)`
C. `2^(2)*3^(3)*7^(2)`
D. `2^(3)*3^(2)*7^(3)`

Answer

C

Explanation

Let middle largest digit be r, then digits available for remaining four places are 0,1,2,3, . . ,r-1.
Number of ways filling remaining four places
`=underset(r=4)overset(9)(sum)(.^(r)P_(4)-.^(r-1)P_(3))=underset(r=4)overset(9)(sum)(r-1)xx.^(r-1)P_(3)`
`=3xx.^(3)P_(3)+4xx.^(4)P_(3)+5xx.^(5)P_(3)+6xx.^(6)P_(3)+7xx.^(7)P_(3)+8xx.^(8)P_(3)`
`=5292=2^(2)*3^(3)*7^(2)`.