Σ_(j=o) ^m ^a C_j ^b C_(m−j) = ^(a+b) C


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Question Number 91230 by waseemyaqoobese@gmail.com last updated on 28/Apr/20

$\underset{j = o}{\sum^{m}} \overset{a}{} C_{j} \overset{b}{} C_{m - j} = \overset{a + b}{} C_{m}$ $s o l v e t h i s p r o b l e m$
	Answered by ~blr237~ last updated on 28/Apr/20

	Answered by mr W last updated on 28/Apr/20

	$w e c a n g e t t h e c o e f . o f x^{m} i n {(1 + x)}^{a + b}$ $i n t w o w a y s :$ ${(1 + x)}^{a + b} = \underset{m = 0}{\sum^{a + b}} C_{m}^{a + b} x^{m}$ $o r$ ${(1 + x)}^{a} {(1 + x)}^{b} = (\underset{j = 0}{\sum^{a}} C_{j}^{a} x^{j}) (\underset{i = 0}{\sum^{b}} C_{i}^{b} x^{i})$ $= \underset{m = 0}{\sum^{a + b}} (\underset{j = 0}{\sum^{m}} C_{j}^{a} x^{j} C_{m - j}^{b} x^{m - j})$ $= \underset{m = 0}{\sum^{a + b}} (\underset{j = 0}{\sum^{m}} C_{j}^{a} C_{m - j}^{b} x^{m})$ $\Rightarrow C_{m}^{a + b} = \underset{j = 0}{\sum^{m}} C_{j}^{a} C_{m - j}^{b}$
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