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Chapter 13: Problem 12

Evaluate each expression. $${ }_{6} P_{0}$$

Short Answer

Expert verified
\(_6 P_0 = 1\)

Step by step solution

01

Understanding Permutations

Permutations are used to determine the number of ways to arrange a subset of items from a larger set. The notation \(_n P_k\) represents the number of permutations of n items taken k at a time.
02

Formula for Permutations

The formula for permutations is given by: \(_n P_k = \frac{n!}{(n-k)!}\). In this exercise, n is 6 and k is 0.
03

Substitute Values

Now, substitute the values of n and k into the formula: \(_6 P_0 = \frac{6!}{(6-0)!} = \frac{6!}{6!}\).
04

Simplify the Fraction

Simplify the fraction: \( \frac{6!}{6!} = 1 \) because any non-zero number divided by itself is 1.

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Key Concepts

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

Permutation Formula
Permutations are a fundamental concept in algebra and combinatorics. They help us understand how to arrange a subset of items from a larger set. The permutation formula is denoted by \(_n P_k\), where \(_{n}\ P_{k}\) represents the number of ways to arrange \('k'\) items from \('n'\) distinct items.

The formula for permutations is: \(_n P_k = \frac{n!}{(n-k)!}\).

This formula tells us how many different ways we can order \(k\) items out of \(n\) items. It involves using factorials to calculate the arrangements.
Factorials in Algebra
Factorials are represented by the symbol \(!\) and are a key part of calculating permutations. A factorial of a non-negative integer \('n'\) is the product of all positive integers less than or equal to \(n\).

For example:

\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]

Factorials grow very quickly with larger numbers.

When calculating permutations, you often use factorials to find total arrangements of a set and to account for the number of ways items can be arranged within subsets.
Arranging Subsets
Arranging subsets deals with how we select and organize items in specific ways. With permutations, this means taking a specific number of items from a larger set and arranging them in different sequences.

Consider a set of \(n\) items. If we want to arrange \(k\) items out of this set, we use the permutation formula: \(_n P_k\).

The notation \( _n P_k \) considers the order of selection. For example, the number of ways to arrange 3 out of 5 items: \[ 5 P 3 = \frac{5!}{(5-3)!} = 5 \times 4 \times 3 = 60\].

This shows that there are 60 possible arrangements when taking 3 items out of 5.

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Most popular questions from this chapter

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