Question

Simplification of Choose Functions

Answer 1

How do I simplify the LHS to be the RHS?

\[\left(\begin{matrix}x \\ n\end{matrix}\right)-\left(\begin{matrix}x-1 \\ n\end{matrix}\right)=\left(\begin{matrix}x-1 \\ n-1\end{matrix}\right)\]
I tried expanding into its factorial equivalents, but can't make progress from there.

Answer 2

is this algebra 2

Answer 3

No, this is probability/combinatorics theory.

Answer 4

Nevermind, it follows from the recursive definition of binomial coefficients. Namely,

\[{n \choose k}={n-1 \choose k-1}+{n-1 \choose k}\quad\text{for all integers n,k>0} \]