Binomial coefficient (case differentiation)

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If you want to proof $\binom{n}{k}$ + $\binom{n}{k+1}$ = $\binom{n+1}{k+1}$, you have to differentiate between 5 cases. Why 5 cases and where do they come from ? It should have something to do with $\binom{N}{k}$ = $\frac{n!}{k!(n-k)!}$ .