Question

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
n · _^n − 1C_{r − 1} = (n − r + 1) ⁿC_{r − 1}

Answer 1

\[LHS = n . {}^{n - 1} C_{r - 1} \]
\[ = \frac{n \left( n - 1 \right)!}{\left( r - 1 \right)! \left( n - 1 - r + 1 \right)!} \]
\[ = \frac{n!}{\left( r - 1 \right)! \left( n - r \right)!}\]
\[RHS = \left( n - r + 1 \right) {}^n C_r \]
\[ = \left( n - r + 1 \right) \frac{n!}{\left( r - 1 \right)! \left( n - r + 1 \right)!} \]
\[ = \left( n - r + 1 \right)\frac{n!}{\left( r - 1 \right)! \left( n - r + 1 \right)\left( n - r \right)!} \]
\[ = \frac{n!}{\left( r - 1 \right)! \left( n - r \right)!}\]

∴ LHS = RHS