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help with combinatorial proofs

pls i need someone to help prove these.

\(proof\) that:

1) \(^{n}C_{r} = ^{n-1}C_{r} +^{n-1}C_{r-1}\).

2) \(\sum {^{k}_{r=0}} ^{m}C_{r} + ^{n}C_{k-r} =^{m+n}C_{k}\).

i'll be grateful if anyone can help with these!!!

Note by Samuel Ayinde
2 years, 1 month ago

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1) \(^{n-1}C_{r}+^{n-1}C_{r-1}\)

\(=\frac{(n-1)!}{r!(n-r-1)!}+\frac{(n-1)!}{(r-1)!(n-r)!}\)

\(=\frac{(n-1)!}{(r-1)!(n-r-1)!}\left(\frac{1}{r}+\frac{1}{n-r}\right)\)

\(=\frac{(n-1)!}{(r-1)!(n-r-1)!}\left(\frac{n}{r(n-r)}\right)\)

\(=\frac{n!}{r!(n-r)!}=^{n}C_{r}\) Omkar Kulkarni · 2 years ago

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2) Make use of this: \(\displaystyle\sum{_{r=0}^{k}}^{n}C_{k-r}=\displaystyle\sum{_{r=0}^{k}}^{n}C_{r}\)

I can't seem to find a solution. Do reply if you manage to prove it! Omkar Kulkarni · 2 years ago

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