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The total number of five-digit numbers o...

The total number of five-digit numbers of different digits in which the digit in the middle is the largest is 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

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)`

Text Solution

Verified by Experts

The correct Answer is:
C
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)`.
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