For any positive integers m-n with n - m- let - n- m - - nCm- Prove that - n- m - - - n-1- m - - - n-2- m - - - - - m- m - - - n-1- m-1 -

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Multiplication of Matrices

For any posit...

Question

For any positive integers m,n $(w i t h n \geq m)$ , let $(\begin{matrix} n m \end{matrix}) =^{n} C_{m}$ . Prove that
$(\begin{matrix} n m \end{matrix}) + (\begin{matrix} n - 1 m \end{matrix}) + (\begin{matrix} n - 2 m \end{matrix}) + . . . . . . . . . . + (\begin{matrix} m m \end{matrix}) = (\begin{matrix} n + 1 m + 1 \end{matrix})$

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m, n (with n \geq m) =^{n} C_{m}

(\begin{matrix} n m \end{matrix}) + (\begin{matrix} n - 1 m \end{matrix}) + (\begin{matrix} n - 2 m \end{matrix}) + . . . + (\begin{matrix} m m \end{matrix}) =

\frac{m (n + 1) + 1}{m (n + 1) - n}

m

n

m, n

n \geq m,

(\begin{matrix} n m \end{matrix}) =^{n} C_{m} .

(\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})

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n^{m} + 1

1 + n + n^{2} . . . . . n^{125}

n (>)

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(n^{m} + 1)

1 + n + n^{2} + . . . n^{225}

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