Write _r=0^m ^n+r C_ r in the simplified form. - Vidyadip

Question

Write $\sum\limits_\text{r=0}^\text{m}\ {^\text{n+r}}\text{C}_{\text{r}}$ in the simplified form.

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Answer

$\sum\limits_\text{r=0}^\text{m}\ {^\text{n+r}}\text{C}_{\text{r}}$
$={^\text{n}}\text{C}_{\text{0}}+{^\text{n+1}}\text{C}_{\text{1}}+{^\text{n+2}}\text{C}_{\text{2}}+{^\text{n+3}}\text{C}_{\text{3}}+ ....+{^\text{n+m}}\text{C}_{\text{m}}$
$=\frac{\text{n}!}{0!\text{n}!}+\frac{(\text{n+1})!}{1!\text{n}!}+\frac{(\text{n}+2)!}{2!\text{n}!}+\frac{(\text{n}+3)!}{3!\text{n}!}+....+\frac{(\text{n+m})!}{\text{n}!\text{m}!}$
$=\frac{(\text{m}!)(\text{n}!)+(\text{n}+1)!\text{m}!+(\text{n+2})!\text{m}!+...+(\text{n}+\text{m})!\text{m}!}{\text{n}!\text{m}!}$
$=\frac{(\text{n}+\text{m}+1)}{\text{m}!(\text{n}+1)!}$
$={^\text{n+m+1}}\text{C}_{\text{n+1}}$

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