Question

Can someone help prove to me Pascal's Identity?

Answer 1

Probably. What method are you trying to use?

Answer 2

Algebraically.

Answer 3

Well, we start out with \[\binom{n}{k}+\binom{n}{k-1}\]and our goal is to show that this is equal to \[\binom{n+1}{k}\]First step, write everything out.

Answer 4

\[\begin{align} \binom{n}{k}+\binom{n}{k-1}&=\frac{n!}{k!(n-k)!}+\frac{n!}{(k-1)!(n-(k-1))!} \\
&=\frac{n!}{k!(n-k)!}+\frac{n!}{(k-1)!(n-k+1)!} \\
&=\frac{n!(n-k+1)}{k!(n-k+1)!}+\frac{kn!}{k!(n-k+1)!}
\end{align}\]

Answer 5

In my humble opinion, many of these proofs just require using the definition.

Answer 6

Ok thanks.

Answer 7

Combine the fractions to get
\[\begin{align}&=\frac{n!(n-k+1)+kn!}{k!(n-k+1)!}\\
&=\frac{n!(n-k+1+k)}{k!(n-k+1)!}\\
&=\frac{n!(n+1)}{k!((n+1)-k)!}
\end{align}\]You should be able to take it from there.