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Prove that ^n-1C_r-1 +^n-1C_r = ⁿC_r

Taking LHS,^{n - 1} C_{r - 1} +^{n - 1} C_{r} = \frac{(n - 1)!}{(r - 1)! \times (n - r)!} + \frac{(n - 1)!}{(r)! \times (n - r - 1)!} = \frac{(n - 1)!}{(r - 1)! \times (n - r) \times (n - r - 1)!} + \frac{(n - 1)!}{r \times (r - 1)! \times (n - r - 1)!} = \frac{(n - 1)!}{(r - 1)! (n - r - 1)!} [\frac{1}{n - r} + \frac{1}{r}] = \frac{(n - 1)!}{(r - 1)! (n - r - 1)!} [\frac{n - r + r}{r (n - r)}] = \frac{(n - 1)!}{(r - 1)! (n - r - 1)!} [\frac{n}{r (n - r)}] = \frac{n (n - 1)!}{r (r - 1)! \times (n - r) \times (n - r - 1)!} = \frac{n!}{r! \times (n - r)!} =^{n} C_{r}

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