Question

Will fan and medal!!!! Please help!!!
Prove that the following relationship is true:
nCr + nCr + 1 = n + 1Cr + 1
Use the least common denominator method.

Answer 1

\[\large\begin{align*}{}_nC_r+{}_nC_{r+1}&=\frac{n!}{r!(n-r)!}+\frac{n!}{(r+1)!(n-r-1)!}\\\\
&=\frac{n!\color{red}{(r+1)}}{\color{red}{(r+1)}r!(n-r)!}+\frac{n!\color{red}{(n-r)}}{(r+1)!\color{red}{(n-r)}(n-r-1)!}\\\\
&=\frac{n!\color{red}{(r+1)}}{(r+1)!(n-r)!}+\frac{n!\color{red}{(n-r)}}{(r+1)!(n-r)!}\\\\
&=\frac{n!\left(r+1+n-r\right)}{(r+1)!(n-r)!}\\\\
&=\frac{n!\left(n+1\right)}{(r+1)!(n-r)!}\\\\
&=\frac{\left(n+1\right)!}{(r+1)!((n+1)-(r+1))!}
\end{align*}\]

Answer 2

@sithsandgiggles THANK YOU SOOOOO MUCHHHH