Problem 59 - 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

The binomial coefficient is calculated as \(\frac{n!}{r!(n-r)!}\). As an example, \(\left(\begin{array}{l}6 \ 2\end{array}\right)\) is calculated as 15.

Step 1: Understand the Concept of Factorial

First, you need to understand the concept of factorial. The factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). It's calculated as \(n*(n-1)*(n-2)*...*3*2*1\). For example, \(5! = 5*4*3*2*1 = 120\). Also, remember that the factorial of 0 is 1.

Step 2: Understand the Binomial Coefficient

Next, you need to understand the binomial coefficient, which is denoted as \(\left(\begin{array}{l}n \ r\end{array}\right)\). This formula represents the number of ways to choose \(r\) elements from a set of \(n\) elements. It's calculated as \(\frac{n!}{r!(n-r)!}\). The positions of the elements are not considered during the selection.

Step 3: Substitute and Calculate

Now, using defined values for \(n\) and \(r\), substitute these into the formula and perform the calculation. \n\nFor example, consider \(\left(\begin{array}{l}6 \ 2\end{array}\right)\): \n\nFirst calculate the factorials: \(6!= 6*5*4*3*2*1=720\), \(2!=2*1=2\) and \((6-2)!=4*3*2*1=24\). \n\nThen calculate the binomial coefficient: \(\frac{720}{2*24} = 15\). So, \(\left(\begin{array}{l}6 \ 2\end{array}\right) = 15\).