Full solution

\left(\begin{array}{c}n+k-1 \\ n-1\end{array}\right)

Identify binomial coefficient notation: Identify the binomial coefficient notation. The binomial coefficient $(n+k-1, n-1)$ is read as $n+k-1$ choose $n-1$ . This represents the number of ways to choose $n-1$ items from a set of $n+k-1$ distinct items without regard to the order of selection.
Use formula: Use the formula for the binomial coefficient. The binomial coefficient can be calculated using the formula: $\newline$ $(a,b) = \frac{a!}{b! \cdot (a-b)!}$ $\newline$ where $a!$ denotes the factorial of $a$ , and $a$ and $b$ are non-negative integers with $a \geq b$ .
Apply formula: Apply the formula to the given binomial coefficient. Substitute $a$ with $n+k-1$ and $b$ with $n-1$ : $\binom{n+k-1}{n-1} = \frac{(n+k-1)!}{(n-1)! * ((n+k-1)-(n-1))!}$
Simplify denominator: Simplify the expression in the denominator. Calculate $((n+k-1)-(n-1))$ : $((n+k-1)-(n-1)) = n+k-1-n+1 = k$
Substitute and simplify: Substitute the simplified expression back into the formula: $\newline$ $[n+k-1],[n-1]$ = $\frac{(n+k-1)!}{(n-1)! \cdot k!}$ $\newline$ This is the simplified form of the binomial coefficient.