1.

The value of the expression `""^(47)C_(4) + sum_(i=1)^(5) ""^(52-i)C_(3)` isA. `""^(47)C_(5)`B. `""^(52)C_(5)`C. `""^(52)C_(4)`D. None of these

Answer» Correct Answer - 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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