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Chapter 11: Problem 40

In how many ways can 7 of 10 rats be arranged in a row for a genetics experiment?

Short Answer

Expert verified
The number of ways to arrange 7 out of 10 rats is 604,800.

Step by step solution

01

Understand the problem

The goal is to determine the number of ways to arrange 7 rats out of 10 in a row. This is a permutation problem because the order in which the rats are arranged matters.
02

Identify the key formula

Use the permutation formula for arranging r items out of n, which is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \]
03

Substitute the values

For this problem, we want to arrange 7 out of 10 rats. Therefore, substitute \( n = 10 \) and \( r = 7 \) into the formula: \[ P(10, 7) = \frac{10!}{(10-7)!} = \frac{10!}{3!} \]
04

Calculate the factorial values

Calculate 10! (10 factorial) and 3! (3 factorial): \[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \] \[ 3! = 3 \times 2 \times 1 = 6 \]
05

Divide to find the permutation

Finally, divide 10! by 3! to find the number of ways to arrange 7 out of 10 rats: \[ P(10, 7) = \frac{3,628,800}{6} = 604,800 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

permutation formula
Permutations help us figure out the number of ways we can arrange a subset of items. They are particularly useful when the sequence matters. For example, arranging books on a shelf or organizing participants in a race. The key formula used for permutations is: \[ P(n, r) = \frac{n!}{(n-r)!} \]
If you're unfamiliar with the symbols:
  • \(P(n, r)\) represents the number of permutations
  • \(n\) is the total number of items available
  • \(r\) is the number of items to arrange

In our example, we have 10 rats but want to arrange only 7. Plugging these values into our formula, we get:
\[ P(10, 7) = \frac{10!}{(10-7)!} = \frac{10!}{3!} \]
This formula only makes sense once you understand factorials, which we鈥檒l explore next.
factorials
Factorials are represented by the exclamation mark (!), and they signify the product of all positive integers up to a given number. It鈥檚 like multiplying a series of numbers in descending order. For instance:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] In our example, we need to calculate \(10!\) and \(3!\). Here鈥檚 how:
  • \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800\)
  • \(3! = 3 \times 2 \times 1 = 6\)

Why do we need these? Because they make it easier to calculate permutations efficiently, avoiding huge numbers! Next, we鈥檒l look at how to arrange items using these results.
arranging items
Arranging items is all about putting things in a specific order. Since order matters in permutations, the sequence affects the count. In our case, we need to arrange 7 rats out of 10. Using our previous results: \[ P(10, 7) = \frac{10!}{3!} = \frac{3,628,800}{6} = 604,800 \]
This means there are 604,800 unique ways to arrange these 7 rats in a row. Here鈥檚 a quick step checklist:
  • Identify total items ()
  • Determine the subset ()
  • Calculate both factorials (! and !-)
  • Substitute these into the permutation formula

By understanding and following these steps, calculating permutations becomes much easier!

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Most popular questions from this chapter

Prove statement for positive integers \(n\) and \(r,\) with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, 1)=n$$

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{5}(5 i+3)$$

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=\frac{1+4 n}{2 n}$$

Find all natural number values for \(n\) for which the given statement is false. $$3^{n}>2 n+1$$

You are offered a 6 -week summer job and are asked to select one of the following salary options. Option I: 5000 dollars for the first day with a 10,000 dollars raise each day for the remaining 29 days (that is, 15,000 dollars for day 2,25,000 dollars for day \(3,\) and so on) Option 2: 0.01 dollars for the first day with the pay doubled each day (that is, 0.02 dollars for day \(2,\) 0.04 dollars for day 3, and so on. Which option would you choose?
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