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I am working through a (rather old) pure maths book as a hobby and have got to a section showing proof of the binomial theorem using induction. Part way through the working the book says:

$_kC_r + _kC_{r-1} = \frac{k!}{r!(k-r)!} + \frac{k!}{(r-1)!(k-\{r-1\})!}$

which I can follow. But I cannot see how the following line is derived:

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

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

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In general $$\frac{a}{bc}+\frac{a}{bd} = \frac{a(d+c)}{bcd}$$ and here

  • $a=k!$,
  • $b=(r-1)!(k-r)!$,
  • $c=r$,
  • $d=k-r+1$.
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