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Chapter 14: Problem 1

Evaluate the given binomial coefficient. $$\left(\begin{array}{l}8 \\\3\end{array}\right)$$

Short Answer

Expert verified
The binomial coefficient \[ {8 \choose 3} \] evaluates to 112.

Step by step solution

01

Identify n and k

In this problem, the binomial coefficient is \[ {8 \choose 3} \] Hence, n = 8 and k = 3.
02

Substitute into the formula

Now we substitute these values into the binomial coefficient formula: \[ {n \choose k} = \frac{n!}{k!(n-k)!} \] resulting in: \[ {8 \choose 3} = \frac{8!}{3!(8-3)!} \]
03

Perform the factorials

Perform factorials for the numbers: \[ = \frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \] Some numbers from the numerator and the denominator can be cancelled out.
04

Cancel out the terms and compute the result

\[ = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} \] Computing this evaluation we get: \[ = 8 \cdot 7 \cdot 2 = 112 \]

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Most popular questions from this chapter

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. Evaluate without using a calculator: \(\frac{600 !}{599 !}\)

Use the formula for the value of an annuity to solve Exercises. Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 75\) at the end of each month in an IRA that pays \(6.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) b. Find the interest.

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