If $\sum_\limits{k=1}^{n-r} { }^{{n}-{k}} {C}_{{r}}={ }^{{x}} {C}_{{y}}$ then

JEE MATHEMATICS MCQ: BINOMIAL THEOREM

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\sum\limits_{{k=1}}^{n-r} { }^{{n}-{k}} {C}_{{r}}={ }^{{x}} {C}_{{y}}

then

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Correct option: B

x=n, y=r+1

Option B (

x=n, y=r+1

) is correct because the sum of the series can be simplified to a single binomial coefficient using the identity

\sum\limits_{{k=1}}^{n-r} { }^{{n}-{k}} {C}_{{r}}={ }^{{n}} {C}_{{r}+1}

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\left({ }^{10} C_0\right)^2-\left({ }^{10} C_1\right)^2+\ldots-\left({ }^{10} C_9\right)^2+\left({ }^{10} C_{10}\right)^2

a

d

\left( {n + 1} \right)

a{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - ... + ...

(1-x)^{201}\left(1+x+x^2\right)^{200}

\left(\frac{1}{x}+2 x\right)^n

{x^5}

\frac{{1 + {x^2}}}{{1 + x}},\left| x \right| < 1,

\left(\begin{array}{c}2 n+1 \\ 0\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 3\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 6\end{array}\right)+\ldots=170

n