Problem 4 - Algebra and Trigonometry

Question

Use the formula for \(_{n} P_{r}\) to evaluate each expression. \(_{10} P_{4}\)

Step-by-Step Solution

Answer

The solution for the given permutation \(_{10} P_{4}\) is 5040.

Step 1: Understanding the permutation formula

Permutation is the way we determine how many ways we can select and arrange r items from n items. The formula for permutation is given by \(_{n}P_{r} = \frac{n!}{(n-r)!}\) where n! refers to the factorial of n, which is the product of all positive integers up to n and (n-r)! is the factorial of (n-r). Here, we are asked to compute \(_{10}P_{4}\), which means that we have 10 items and we are to determine how many ways we can select and arrange 4 of them.

Step 2: Substituting into the permutation formula

Substitute n=10 and r=4 into the formula to get \(_{10}P_{4} = \frac{10!}{(10-4)!}\). Such that the formula now reads \(_{10}P_{4} = \frac{10!}{6!}\).

Step 3: Calculation of factorials

The next step is to evaluate the factorials (10! and 6!) . Calculate 10! as (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) and 6! as (6 * 5 * 4 * 3 * 2 * 1). Now, \(_{10}P_{4}\) is represented as follows \(_{10}P_{4} = \frac{10!}{6!} = \frac{10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1}{6 * 5 * 4 * 3 * 2 * 1}\).

Step 4: Simplify the Expression

Observe that the factorials in the numerator and the denominator cancel out up to 6 in the numerator and we are left with 10 * 9 * 8 * 7 = 5040.

Key Concepts

Factorials

Factorials are an essential concept in mathematics, especially within the realms of permutations and combinations. The factorial of a number, denoted by !, is the product of all positive integers up to that number. For instance, 10! (read as '10 factorial') is calculated as:

• 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

Understanding factorials is crucial because they are the backbone of many formulas in combinatorics. They help determine the number of ways to arrange items. In permutations, factorial calculations often simplify expressions by canceling out common terms between numerator and denominator.
Whenever dealing with these mathematical problems, recalling the fundamental operation of a factorial will aid you in making accurate calculations.

Permutation Formula

The permutation formula \(_{n}P_{r} = \frac{n!}{(n-r)!}\) helps us ascertain the number of ways to arrange \(r\) items out of \(n\) distinct items. The arrangement or ordering of these items is crucial and therefore makes permutations distinct from other combinations.
Let's break down what's happening in the formula:

• \(n!\) is the factorial of the total number of items, giving the total ways to arrange them.
• \((n-r)!\) accounts for the elements that are left out, effectively reducing the total number of arrangements.

For example, evaluating \(_{10}P_{4}\):

• \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)
• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\)

Canceling 6! from both, we perform \(10 \times 9 \times 8 \times 7 = 5,040\). This formula efficiently provides the solution to permutation problems without manually listing out all possibilities.

Combinatorics

Combinatorics is a fascinating area of mathematics focusing on counting, arrangement, and combination of elements within a set. It encompasses both permutations and combinations, with permutations emphasizing ordered arrangements.
Understanding the difference between permutations and combinations is vital. While permutations consider order important (like arranging books on a shelf), combinations care only about selection (such as choosing books to borrow, ignoring their order).
Tools like factorials and the permutation formula are fundamental in combinatorics. These tools help solve problems involving:

• Specific arrangement of elements (permutations).
• General selection of items (combinations).
• Counting principle applications.

In practice, combinatorics is widely used in fields such as computer science, cryptography, and optimization, making it a crucial skill for anyone tackling mathematical problems or puzzles.