Problem 58 - College Algebra

Question

Explain how to evaluate \(\left(\begin{array}{l}n \\ r\end{array}\right) .\) Provide an example with your explanation.

Step-by-Step Solution

Answer

To calculate a binomial coefficient, use the formula \(\left(\begin{array}{l}n \ r\end{array}\right) = \frac{n!}{r!(n-r)!}\). For example, \(\left(\begin{array}{l}5 \ 3\end{array}\right) = \frac{5!}{3!(5-3)!} = 10 \).

Step 1: Understanding Factorials

A factorial is a function that multiplies a given number by every number below it until 1. This is denoted as n!. For example, the factorial of 5 (written as 5!) would be calculated as 5 * 4 * 3 * 2 * 1 = 120.

Step 2: Binomial Coefficient Formula

The formula for calculating binomial coefficients is \(\left(\begin{array}{l}n \ r\end{array}\right) = \frac{n!}{r!(n-r)!}\). This means that to evaluate a binomial coefficient, we need to calculate the factorial of n, the factorial of r, and the factorial of (n-r), and then conduct the division as stated in the formula.

Step 3: Example

Now let's calculate the binomial coefficient \(\left(\begin{array}{l}5 \ 3\end{array}\right)\). Applying the formula, we find: \(\left(\begin{array}{l}5 \ 3\end{array}\right) = \frac{5!}{3!(5-3)!} = \frac{120}{6*2} = 10\)