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The value of ((50),(6))-((5),(1))((40),(...

The value of `((50),(6))-((5),(1))((40),(6))+((5)/(2))((30),(6))-((5),(3))((20),(6))+((5),(4))((10),(6))` where `((n),(r ))` denotes `"^(n)C_(r )`, is

A

`15625`

B

`0`

C

`1000000`

D

`2250000`

Text Solution

Verified by Experts

The correct Answer is:
D
`(d)` `'^(50)C_(6)-^(5)C_(1)^(40)C_(6)+^(5)C_(2)^(30)C_(6)-^(5)C_(3)^(20)C_(6)+^(5)C_(4)^(10)C_(6)`
`="coefficient of" x^(6) "in" ['^(5)C_(0)(1+x)^(50)-^(5)C_(1)(1+x)^(40)+^(5)C_(2)(1+x)^(30)-^(5)C_(3)(1+x)^(20)+^(5)C_(4)(1+x)^(10)-^(5)C_(5)(1+x)^(0)]`
`="coefficient" x^(6) "in" [(1+x)^(10)-1]^(5)`
`="coefficient" x^(6) "in" ('^(10)C_(1)x+^(10)C_(2)x^(2)+....)^(5)`
`="coefficient" x "in" ('^(10)C_(1)x+^(10)C_(2)x+....)^(5)`
`='^(5)C_(1)('^(10)C_(2))('^(10)C_(1))^(4)=2250000`
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