Prove that -n-C-r- - -n - 1-C-r- - -n - 2-C-r- - - - -r-C-r- - -n - 1-C-r - 1-

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To prove-nCr-n-x2212-1Cr-n-x2212-2Cr-rCr-n-1Cr-1nCr-n-x2212-1Cr-n-x2212-2Cr-r-1Cr-rCr-n-1Cr-1-x2235-nCn-n-1Cn-1-x21D2-nCr-n-x2212-1Cr-n-x2212-2Cr-r-2Cr-r-1Cr-r-1Cr-1-x21D2-r-2Cr-1-We know that n-1Cr-1-nCr-1-nCr-x2234-nCr-n-x2212-1Cr-n-x2212-2Cr-r-2Cr-r-2Cr-1-n-1Cr-1-x21D2-nCr-n-x2212-1Cr-n-x2212-2Cr-r-3Cr-1-n-1Cr-1By continuing this- we get-x21D2-nCr-n-x2212-1Cr-n-x2212-1Cr-1-n-1Cr-1-x21D2-nCr-nCr-1-n-1Cr-1-x21D2-n-1Cr-1-n-1Cr-1L-H-S-R-H-S-Hence proved-

Prove that nCr+n−1Cr+n−2Cr+.....+rCr=n+1Cr+1.

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