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Chapter 8: Problem 58

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

Example

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

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