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. none of these

Options

A. `30xx30!`
B. `33xx3!`
C. `sum_(n=4)^(9)""^(n)C_(4)xx4!`
D. none of these

Explanation

The largest digit in the middle of the five digit number can be 4 (when numbers are formed with digits 0, 1, 2, 3, 4), 5 (when 0, 1, 2, 3, 4, 5 are used),…..,9 (when 0, 1, 2,……,9 are used).
`:.` Number of five digit numbers with largest digit 5 in the middle
`=(""^(4)C_(4)xx4!-""^(3)C_(3)xx3!)`
Number of five digit numbers with lardest digit 5 in the middle
`=(""^(5)C_(4)xx4!-""^(3)C_(3)xx3!)`
Hence, required number of numbers
`=(""^(4)C_(4)xx4!-""^(3)C_(3)xx3!)+(""^(5)C_(4)xx4!-""^(4)C_(3)xx3!)`
`+(""^(6)C_(4)xx4!-""^(5)C_(3)xx3!)+.....+(""^(9)C_(4)+4!-""^(8)C_(3)xx3!)`