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If N is a prime number which divides S=^...

If `N` is a prime number which divides `S=^(39)P_(19)+^(38)P_(19)+^(37)P_(19)+…+"^(20)P_(19)`, then the largest possible value of `N` among following is

A

`41`

B

`31`

C

`37`

D

`19`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `S=^(39)P_(19)+^(38)P_(19)+^(37)P_(19)+....+^(20)P_(19)`
`=19!{'^(30)C_(19)+^(38)C_(19)+^(37)C_(19)+....+^(20)C_(19)}`
`=19!{'^(39)C_(19)+^(38)C_(19)+^(37)C_(19)+....+('^(20)C_(19)+^(20)C_(19))-1}`
`=19!{'^(39)C_(19)+^(38)C_(19)+....+('^(21)C_(19)+^(21)C_(19))-1}`
`=19!{'^(40)C_(20)-1}`
`=((41-1)(41-2)(41-3).....(41-20)-20!)/(20)=(41k)/(20)`
`implies` greatest prime number is `41`
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