Prove that:

n! / r! x (n-r)! + n! / (r-1)! x (n-r+1) = (n+1)! / r! x (n-r+1)!

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Solution

LHS=n!/[r!(n-r)!]+n!/[(r-1)!(n-r+1)!] =n!/[r(r-1)!(n-r)!]+ n!/[(r-1)!(n-r+1)(n-r)!]] =n!/[(r-1)!(n-r)!]{[1/r]+[1/(n-r+1)]} =n!/[(r-1)!(n-r)!]{(n+1)/r(n-r+1)} =(n+1)n!/[r(r-1)!(n-r+1)(n-r)!] =(n+1)!/[r!(n-r+1)!] =n+1Cr

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Similar questions

\sum_{r = 0}^{n} {r + 1}^{n} C_{r} = (n + 2) 2^{n - 1}

\sum_{r = 0}^{n} {r + 1}^{n} C_{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1} .

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1} .

n \sum r = 0 (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}

n \sum r = 0 (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . + C_{n} x^{n}

C_{0} C_{r} + C_{1} C_{r + 1} + . . . . . . + C_{n - r} C_{n} = \frac{(2 n)!}{(n + r)! (n - r)!}

f (x) = x^{n}

f (1) + \frac{f^{'} (1)}{1} + \frac{f^{''} (1)}{2!} + \frac{f^{'''} (1)}{3!} + . . . + \frac{^{n} f (1)}{n!} = 2^{n}

{(1 + x)}^{n} = n \sum r = 0^{n} C_{r} x^{r}

n \sum r = 0^{n} C_{r} = 2^{n}

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