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

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

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

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

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

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! · 2 years ago