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Chapter 14: Problem 61

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

Short Answer

Expert verified
Firstly, understand the notation \( \left(\begin{array}{l}n \ r\end{array}\right) \) as a combination. It is calculated by the formula \( \left(\begin{array}{l}n \ r\end{array}\right) = \frac{n!}{r!(n-r)!} \), where \( ! \) denotes a factorial. As an example, \( \left(\begin{array}{l}5 \ 3\end{array}\right) = 10 \).

Step by step solution

01

Understand the notation

The symbol \( \left(\begin{array}{l}n \ r\end{array}\right) \) represents a combination, which denotes the number of ways we can choose \( r \) objects from \( n \) without regard to the order of selection.
02

Apply the formula

The formula used to calculate a combination is \( \left(\begin{array}{l}n \ r\end{array}\right) = \frac{n!}{r!(n-r)!} \). The exclamation point denotes factorial, which means the product of an integer and all the integers below it, down to 1.
03

Calculate Factorial

Calculate the factorials in the formula. As an example, if we chose \( n = 5 \) and \( r = 3 \), then we would first calculate the factorials: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120, 3! = 3 \times 2 \times 1 = 6 \), and \( (5-3)! = 2! = 2 \times 1 = 2 \)
04

Substitute in the formula

Next, substitute the factorials into the formula and simplifying the fraction. Using our example, we will have \( \left(\begin{array}{l}5 \ 3\end{array}\right) = \frac{5!}{3!(5-3)!} = \frac{120}{6 \times 2} = 10 \)

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Simplify: \(\sqrt{28}-3 \sqrt{7}+\sqrt{63}\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The sequence for the number of seats per row in our movie theater as the rows move toward the back is arithmetic with \(d=1\) so people don't block the view of those in the row behind them.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I was able to find the sum of the first fifty terms of an arithmetic sequence even though I did not identify every term.

Use the formula for the sum of the first n terms of a geometric sequence to solve. A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. \begin{aligned}&16, \quad\quad\quad\quad\quad 0.96(16), \quad(0.96)^{2}(16), \quad(0.96)^{3}(16), \ldots\\\&\begin{array}{|l|l|l|}\hline \begin{array}{l}\text { 1st swing } \\\\\end{array} & \begin{array}{l}\text{ 2nd swing } \\\\\end{array} & \begin{array}{l}\text { 3rd swing } \\\\\end{array} & \begin{array}{l}\text { 4th swing } \\\\\end{array} \\\\\hline\end{array}\end{aligned} After 10 swings, what is the total length of the distance the pendulum has swung? Round to the nearest hundredth of an inch.
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