Prove that i) nCr−1+nCr=n+1Crii) Prove that nCr−2+2.nCr−1+ nCr= n+2Criii) For 3≤r≤n. Prove that n−1Cr+3(n−3Cr−1)+3(n−3Cr−2)+n−3Cr−3

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Prove that i) $^{n} C_{r - 1} +^{n} C_{r} =^{n + 1} C_{r}$
ii) Prove that $^{n} C_{r - 2} + {2.}^{n} C_{r - 1} +^{n} C_{r} =^{n + 2} C_{r}$
iii) For $3 \leq r \leq n .$ Prove that $^{n - 1} C_{r} + 3 (^{n - 3} C_{r - 1}) + 3 (^{n - 3} C_{r - 2}) +^{n - 3} C_{r - 3}$

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i) L HS $^{n} C_{r - 1} +^{n} C_{r}$

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

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

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

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

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

$= \frac{(n + 1)!}{(n - r + 1)! r!} =^{n + 1} C_{r} = R H S$

$L H S =^{n} C_{r - 2} + {2.}^{n} C_{r - 1} +^{n} C_{r}$

$= (^{n} C_{r - 2} +^{n} C_{r - 1}) + (^{n} C_{r - 1} +^{n} C_{r})$

$=^{n + 1} C_{r - 1} +^{n + 1} C_{r} =^{n + 2} C_{r}$

$∴^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

$^{n - 3} C_{r} + 3 (^{n - 3} C_{r - 1}) + 3 (^{n - 3} C_{r - 2}) +^{n - 3} C_{r - 3}$

$= (^{n - 3} C_{r} +^{n - 3} C_{r - 1}) + 2 (^{n - 3} C_{r - 1} +^{n - 3} C_{r - 2}) + (^{n - 3} C_{r - 2} +^{n - 3} C_{r - 3})$

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

$=^{n - 2} C_{r} + 2 (^{n - 2} C_{r - 1}) +^{n - 2} C_{r - 2}$

$= (^{n - 2} C_{r} +^{n - 2} C_{r - 1}) + (^{n - 2} C_{r - 1} +^{n - 2} C_{r - 2})$

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

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