# Problem following working in binomial theorem proof by induction

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:

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

$$\frac{k!}{r!(k-r)!} + \frac{k!}{(r-1)!(k-\{r-1\})!}=\frac{k!(k-r+1)}{r!(k-r)!(k-r+1)} + \frac{k!}{(r-1)!(k-r+1)!}=\frac{k!(k-r+1)}{r!(k-r+1)!} + \frac{k!\cdot r}{r\cdot (r-1)!(k-r+1)!}=\frac{k!(k-r+1)}{r!(k-r+1)!} + \frac{k!\cdot r}{r!(k-r+1)!}=\frac{k!}{r!(k-r+1)!}\{(k-r+1)+r\}$$
In general $$\frac{a}{bc}+\frac{a}{bd} = \frac{a(d+c)}{bcd}$$ and here
• $$a=k!$$,
• $$b=(r-1)!(k-r)!$$,
• $$c=r$$,
• $$d=k-r+1$$.