mCr+ mCr−1⋅ nC1+ mCr−2⋅ nC2+⋯+ nCr=

G.E. = Coefficient of xr in [∑r=0nnCr⋅xr] [∑r=0mmCr⋅xr]= Coefficient of xr in (1+x)m⋅(1+x)n= Coefficient of xr in (1+x)m+n= (m+n)Cr.

mCr+ mCr−1⋅ nC1+ mCr−2⋅ nC2+⋯+ nCr=

G.E. = Coefficient of xr in [∑r=0nnCr⋅xr] [∑r=0mmCr⋅xr]= Coefficient of xr in (1+x)m⋅(1+x)n= Coefficient of xr in (1+x)m+n= (m+n)Cr.

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$^{m} C_{r} +^{m} C_{r - 1} \cdot^{n} C_{1} +^{m} C_{r - 2} \cdot^{n} C_{2} + \dots +^{n} C_{r} =$

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$^{m n} C_{r}$

$^{(m + n)} C_{r + 1}$

$^{(m + n)} C_{r - 1}$

$^{(m + n)} C_{r}$

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G.E. $=$ Coefficient of $x^{r} in [{\sum_{r = 0}^{n}}^{n} C_{r} \cdot x^{r}] [{\sum_{r = 0}^{m}}^{m} C_{r} \cdot x^{r}]$

$=$ Coefficient of $x^{r} in {(1 + x)}^{m} \cdot {(1 + x)}^{n}$

$=$ Coefficient of $x^{r} in {(1 + x)}^{m + n} =^{(m + n)} C_{r}$ .

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mCr+ mCr−1⋅ nC1+ mCr−2⋅ nC2+⋯+ nCr=

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$^{m} C_{r} +^{m} C_{r - 1} \cdot^{n} C_{1} +^{m} C_{r - 2} \cdot^{n} C_{2} + \dots +^{n} C_{r} =$