Question

In Exercises 71-74, evaluate the binomial coefficient using the formula \(\left(\begin{array}{l}k \\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n !}\) where \(k\) is a real number, \(n\) is a positive integer, and \(\left(\begin{array}{l}k \\ 0\end{array}\right)=1\). $$ \left(\begin{array}{l} 5 \\ 3 \end{array}\right) $$

Short answer

The value of the binomial coefficient \(\left(\begin{array}{l}5 \\ 3\end{array}\right)\) is 10.

Step-by-step solution

Identify k and n values

In the binomial coefficient \(\left(\begin{array}{l}5 \\ 3\end{array}\right)\), the value of k is 5 and the value of n is 3.

Apply the binomial coefficient formula

Plug the values of k and n into the binomial coefficient formula: \(\left(\begin{array}{l}k \\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n!}\). Thus, the expression becomes \(\frac{5(5-1)(5-2)}{3!}\).

Simplify the expression

Simplify the expression. The numerator becomes \(5*4*3\) and the denominator becomes \(3*2*1\). So, \(\frac{5(5-1)(5-2)}{3!}\) simplifies to \(\frac{60}{6}\).

Final Calculation

The final calculation is \(\frac{60}{6}\) which equals to 10.