To find the value of \( n \cdot {}^{(n-1)}C_{(r-1)} \), we can proceed with the following steps: 1. Express the Combination: The combination \( {}^{(n-1)}C_{(r-1)} \) is given by: \[ {}^{(n-1)}C_{(r-1)} = \frac{(n-1)!}{(r-1)! \cdot (n-r)!} \] 2. Multiply by \( n \): Multiply the combination by \( n \): \[ n \cdot {}^{(n-1)}C_{(r-1)} = n \cdot \frac{(n-1)!}{(r-1)! \cdot (n-r)!} \] 3. Simplify the Expression: Notice that \( n \cdot (n-1)! = n! \): \[ n \cdot {}^{(n-1)}C_{(r-1)} = \frac{n!}{(r-1)! \cdot (n-r)!} \] 4. Express in Terms of \( nC_r \): The combination \( nC_r \) is given by: \[ nC_r = \frac{n!}{r! \cdot (n-r)!} \] Notice that: \[ \frac{n!}{(r-1)! \cdot (n-r)!} = r \cdot \frac{n!}{r! \cdot (n-r)!} = r \cdot nC_r \] Therefore, the value of \( n \cdot {}^{(n-1)}C_{(r-1)} \) is: \[ \boxed{r \cdot nC_r} \]