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Chapter 11: Problem 59

A group of campers is going to occupy five campsites at a campground. There are 12 campsites from which to choose. In how many ways can the campsites be chosen?

Short Answer

Expert verified
There are 792 ways for the campers to select 5 campsites out of a total of 12.

Step by step solution

01

Identify n and k from the problem

From the problem, the total number of campsites available - the set from which we need to choose - is 12. This is our \( n \) value. The number of campsites we need to select is 5. This is our \( k \) value. So, \( n = 12 \), \( k = 5 \).
02

Apply the formula for combinations

The formula for calculating combinations is \( C(n, k) = \frac{n!}{k!(n - k)!} \). Let's insert our values of \( n \) and \( k \) into it: \( C(12, 5) = \frac{12!}{5!(12 - 5)!} = \frac{12!}{5! \cdot 7!} \).
03

Calculate the factorials

Factorial, denoted with \( ! \), is the product of all positive integers up to that number. Or formally, \( n! = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1 \). We need to calculate \( 12! \) , \( 5! \) and \( 7! \). We have \( 12! = 479001600 \), \( 5! = 120 \), and \( 7! = 5040 \). Let's put these values back into our combination formula: \( C(12, 5) = \frac{479001600}{120 \cdot 5040} \).
04

Complete the calculation

Now we can simply carry out the division to get our final answer. So the calculation is \( C(12, 5) = \frac{479001600}{120 \cdot 5040} = 792 \). So, there are 792 different ways in which the campers can choose 5 campsites out of the available 12.

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Most popular questions from this chapter

We return to our box of chocolates. There are 30 chocolates in the box, all identically shaped. Five are filled with coconut, 10 with caramel, and 15 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting a coconut-filled chocolate followed by a caramel-filled chocolate.

An ice cream store sells two drinks (sodas or milk shakes), in four sizes (small, medium, large, or jumbo), and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?

Evaluate each factorial expression. \(\frac{29 !}{25 !}\)

Explain the Fundamental Counting Principle.

In the original plan for area codes in 1945 , the first digit could be any number from 2 through 9 , the second digit was either 0 or 1 , and the third digit could be any number except 0 . With this plan, how many different area codes are possible?
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