JEE MATHEMATICS MCQ: BINOMIAL THEOREM

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Analyzing +4 | -1

The value of

{ }^n C_0 \cdot{ }^{2 n} C_r-{ }^n C_1,{ }^{2 n-2} C_r

+{ }^n C_2 \cdot{ }^{2 n-4} C_r+\ldots+(-1)^n{ }^n C_n{ }^{2 n-2 n} C_n

View answer and explanation

Correct option: A

\begin{cases}2^{2 n-r}{ }^n C_{r-n}, & \text { if } r>n \\ 0, & \text { if } r < n\end{cases}

Option A is correct because the expression simplifies to the coefficient of

x^{r-n}

(2x+x^2)^n

, which is

2^{2n-r} { }^n C_{r-n}

r>n

and 0 if

r < n

More Questions from BINOMIAL THEOREM

Practice All Chapter MCQs →

\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