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The value of the expression ""^(47)C(4) ...

The value of the expression `""^(47)C_(4) + sum_(i=1)^(5) ""^(52-i)C_(3)` is

A

`""^(47)C_(5)`

B

`""^(52)C_(5)`

C

`""^(52)C_(4)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
Here, `""^(47)C_(4) + overset(5)underset(j = 1)sum ""^(52 - j)C_(3)`
`= ""^(47)C_(4) + ""^(51)C_(3) + ""^(50)C_(3) + ""^(49)C_(3) + ""^(48)C_(3) + ""^(47)C_(3)`
`= (""^(47)C_(4) + ""^(47)C_(3)) + ""^(48)C_(3) + ""^(49)C_(3) + ""^(50)C_(3) + ""^(51)C_(3)`
[using `""^(n)C_(r) + ""^(n)C_(r -1) = ""^(n + 1)C_(r)`]
`= (""^(48)C_(4) + ""^(48)C_(3)) + ""^(49)C_(3) + ""^(50)C_(3) + ""^(51)C_(3)`
`= (""^(49)C_(4) + ""^(49)C_(3)) + ""^(50)C_(3) + ""^(51)C_(3)`
`= (""^(50)C_(4) + ""^(50)C_(3)) + ""^(51)C_(3)`
`= ""^(51)C_(4) + ""^(51)C_(3) = ""^(52)C_(4)`
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