Question

Prove, algebraically and logically, that the combination equation:
nCr + nC(r+1) = (n+1)C(r+1)

Answer 1

\[\left(\begin{matrix}n \\ r\end{matrix}\right) + \left(\begin{matrix}n \\ r+1\end{matrix}\right) = \left(\begin{matrix}n+1 \\ r+1\end{matrix}\right)\]

Answer 2

Do you mean algebraic and combinatorial?

Answer 3

Just without the use of number substitutions.

Answer 4

Why don't you just expand it and see ...

Answer 5

How are you meant to expand it. My teacher showed me very quickly today then rubbed it out, all I remember is it started by changing it to factorial form.

Answer 6

\( \huge \binom{n}{r} = \frac{n!}{r!\times(n-r)!} \)