1.

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

Answer»

`41`
`31`
`37`
`19`

Solution :`(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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