For any positive integer m-n with n- m- let nm - nCm- Thennm- n-1m- n-2m- mm is equal to

We know that, ^nC_r= Coefficient of x^r in (1+x)^n ∴( n m )+ ( n 1 m )+ ( n 2 m )+⋯+ ( m m ) = Coefficient of x^m in (1+x)^n+ Coefficient of x^m in (1+x)^ ...

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Standard XI

Mathematics

Sum of product of binomial coefficients

For any posit...

Question

For any positive integer $m, n$ with $n \geq m,$ let $(\begin{matrix} n m \end{matrix}) =^{n} C_{m} .$ Then
$(\begin{matrix} n m \end{matrix}) + (\begin{matrix} n - 1 m \end{matrix}) + (\begin{matrix} n - 2 m \end{matrix}) + \dots + (\begin{matrix} m m \end{matrix})$ is equal to

^{n} C_{m}

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^{n + 1} C_{m}

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^{n} C_{m + 1}

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^{n + 1} C_{m + 1}

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Solution

The correct option is D $^{n + 1} C_{m + 1}$
We know that,
$^{n} C_{r} =$ Coefficient of $x^{r}$ in $(1 + x)^{n}$
$∴ (\begin{matrix} n m \end{matrix}) + (\begin{matrix} n - 1 m \end{matrix}) + (\begin{matrix} n - 2 m \end{matrix}) + \dots + (\begin{matrix} m m \end{matrix})$
$= Coefficient of x^{m} in (1 + x)^{n} + Coefficient of x^{m} in (1 + x)^{n - 1}$ $+ \dots + Coefficient of x^{m} in (1 + x)^{m}$
$= Coefficient of x^{m} in [(1 + x)^{n} + (1 + x)^{n - 1} + \dots + (1 + x)^{m}]$
$= Coefficient of x^{m} in (1 + x)^{m} {\frac{(1 + x)^{n - m + 1} - 1}{(1 + x) - 1}}$
$= Coefficient of x^{m + 1} in [(1 + x)^{n + 1} - (1 + x)^{m}]$
$= Coefficient of x^{m + 1} in (1 + x)^{n + 1}$
$=^{n + 1} C_{m + 1}$

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Sum of product of binomial coefficients

Standard XI Mathematics

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For any positive integer m,n with n≥m, let (nm)=nCm. Then (nm)+(n−1m)+(n−2m)+⋯+(mm) is equal to

For any positive integer $m, n$ with $n \geq m,$ let $(\begin{matrix} n m \end{matrix}) =^{n} C_{m} .$ Then
$(\begin{matrix} n m \end{matrix}) + (\begin{matrix} n - 1 m \end{matrix}) + (\begin{matrix} n - 2 m \end{matrix}) + \dots + (\begin{matrix} m m \end{matrix})$ is equal to