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Chapter 9: Problem 52

Evaluate \(_{n} C_{r}\) using the formula from this section. $$_{6} C_{3}$$

Short Answer

Expert verified
The value of \(_{6}C_{3} = 20\).

Step by step solution

01

Identify values in the combination formula

In this case, the total number of items, \( n = 6 \), and the chosen number of items, \( r = 3 \).
02

Calculate the factorials

First calculate \( 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 \), then calculate \( 3! = 3 * 2 * 1 = 6 \), and finally calculate \( (6-3)! = 3! = 6 \).
03

Insert the values into the formula

Substitute the calculated values into the combination formula: \( _{6}C_{3} = \frac{6!}{3!(6-3)!} = \frac{720}{6*6} \).
04

Simplify the equation

Solve the equation to get the final result: \( _{6}C_{3} = \frac{720}{36} = 20 \).

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Most popular questions from this chapter

Write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\) th term of the sequence as a function of \(n .\) (Assume \(n\) begins with 1.) $$a_{1}=6, a_{k+1}=a_{k}+2$$

Find the binomial coefficient. \(_{18} C_{2}\)

Write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\) th term of the sequence as a function of \(n .\) (Assume \(n\) begins with 1.) $$a_{1}=14, a_{k+1}=-2 a_{k}$$

Find the binomial coefficient. \(\left(\begin{array}{c}10 \\ 4\end{array}\right)\)

Use a graphing utility to find the partial sum. $$\sum_{j=1}^{200}(10.5+0.025 j)$$
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