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Chapter 11: Problem 23

Use a calculator to evaluate expression. \(_{15} P_{8}\)

Short Answer

Expert verified
259459200

Step by step solution

01

- Understand the Notation

The notation \(_{n} P_{r}\) represents a permutation. This formula is used to calculate the number of ways to arrange r items out of n items. The formula for permutations is given by: \[_{n}P_{r} = \frac{n!}{(n-r)!}\]
02

- Substitute the Values

Substitute the given values into the formula. Here, n = 15 and r = 8. So we need to calculate: \[_{15}P_{8} = \frac{15!}{(15-8)!} = \frac{15!}{7!}\]
03

- Calculate Factorials

Calculate 15! (15 factorial) and 7! (7 factorial). Recall that n! is the product of all positive integers up to n.15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 17! = 7 x 6 x 5 x 4 x 3 x 2 x 1
04

- Simplify the Factorials

Divide 15! by 7! to simplify the expression. Notice that the 7! in the denominator cancels out the 7! portion in the numerator:\[_{15}P_{8} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7!}{7!} = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8\]
05

- Perform the Multiplication

Finally, perform the multiplication:\[15 \times 14 = 210 \]\[210 \times 13 = 2730 \]\[2730 \times 12 = 32760 \]\[32760 \times 11 = 360360 \]\[360360 \times 10 = 3603600 \]\[3603600 \times 9 = 32432400 \]\[32432400 \times 8 = 259459200\]

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Key Concepts

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

Factorials
In mathematics, a factorial, represented by an exclamation mark (!), is the product of all positive integers from 1 up to a given number. For instance, the factorial of 5, denoted as 5!, is calculated as:
5! = 5 脳 4 脳 3 脳 2 脳 1 = 120.
Factorials are foundational in permutations and combinations. They help in counting the number of ways objects can be arranged or chosen. Always remember that 0! is defined to be 1 by convention.
When working with large numbers, such as 15! in permutations, it's important to know the properties of factorials. Factorials can grow very fast, hence using a calculator or factorial table can help simplify and expedite calculations.
Permutation Formula
Permutations are a way of determining the different possible arrangements of a given set of items. The permutation formula, represented as \(_{n} P_{r}\), is used to calculate the number of ways to arrange r items out of n items. The formula is:
\[_{n}P_{r} = \frac{n!}{(n-r)!}\]
For example, in the given problem where we need to find \(_{15} P_{8}\), we can substitute the values accordingly:
  • n = 15
  • r = 8
So, we need to compute: \(_{15}P_{8} = \frac{15!}{(15-8)!} = \frac{15!}{7!}\).
Understanding this formula is crucial for solving permutation-related problems, as it helps in reducing complex arrangements into manageable calculations.
Arrangements
Arrangements or permutations involve the order of items being important. For example, the arrangement of 3 out of 4 books is different with each permutation. The sequence matters in permutations, which sets them apart from combinations.
To understand arrangements better, consider a smaller example with 3 items: A, B, and C. The possible permutations or arrangements of 2 items chosen from these 3 are:
  • AB
  • BA
  • AC
  • CA
  • BC
  • CB
There are 6 distinct ways to arrange 2 items from 3 items. Extending this idea, we see that permutations help us understand how to order a subset of a larger set. The calculated value helps in solving problems related to possible ways to organize or order in everyday scenarios and complex mathematical problems.

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Most popular questions from this chapter

Consider the word BRUCE. (a) In how many ways can all the letters of the word BRUCE be arranged? (b) In how many ways can the first 3 letters of the word BRUCE be arranged?

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$2+4+8+\dots+2^{n}=2^{n+1}-2$$

Prove statement for positive integers \(n\) and \(r,\) with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$\left(\begin{array}{c}n \\\n-r\end{array}\right)=\left(\begin{array}{l}n \\\r\end{array}\right)$$

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{6}\left(2+i-i^{2}\right)$$

It can be shown that $$(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\cdots$$ is true for any real number \(n\) (not just positive integer values) and any real number \(x\), where \(|x| < 1 .\) Use this series to approximate the given number to the nearest thousandth. $$(1.03)^{2}$$
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