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Chapter 11: Problem 18

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{6} C_{0}\)

Short Answer

Expert verified
The value of \({ }_{6} C_{0}\) is 1.

Step by step solution

01

Understanding Combinations

Combinations are a way to calculate the total outcomes of a situation where the order of the outcomes does not matter. More specifically, the formula for combinations is given by \({ }_{n} C_{r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, \(r\) is the number of items to choose, and \(n!\) and \(r!\) denote the factorial of \(n\) and \(r\), respectively.
02

Applying the Combinations Formula

Now, apply the formula to the given expression, \({ }_{6} C_{0}\). The number of total items (\(n\)) is 6 and the number of items to choose (\(r\)) is 0. Plugging these values into the formula gives us: \[C = \frac{6!}{0!(6-0)!}\].
03

Evaluating Factorials

Recall that by definition, 0! is equal to 1. Therefore, the expression simplifies to: \[C = \frac{6!}{1!(6)!}\].
04

Simplify the Expression

Simplify the expression further by calculating the factorials: \[C = \frac{720}{1*720} = 1\]

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Most popular questions from this chapter

The tables in Exercises 3-4 show claims and their probabilities for an insurance company. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it breaks even on its claim costs? c. How much should the company charge to make a profit of \(\$ 50\) per policy? PROBABILITIES FOR HOMEOWNERS' INSURANCE CLAIMS $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Amount of Claim (to the } \\ \text { nearest } \$ \mathbf{\$ 5 0 , 0 0 0 )} \end{array} & \text { Probability } \\ \hline \$ 0 & 0.65 \\ \hline \$ 50,000 & 0.20 \\ \hline \$ 100,000 & 0.10 \\ \hline \$ 150,000 & 0.03 \\ \hline \$ 200,000 & 0.01 \\ \hline \$ 250,000 & 0.01 \\ \hline \end{array} $$

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License plates in a particular state display two letters followed by three numbers, such as AT- 887 or BB-013. How many different license plates can be manufactured for this state?

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