Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{7} P_{4}$$

Step-by-Step Solution

Answer

The number of permutations, \(_{7} P_{4}\), is 840.

Step 1: Identify Variables

From the exercise, we can see that in the symbol \(_{n} P_{r}\), \(n = 7\) represents the total number of items available, while \(r = 4\) stands for the number of items to choose and arrange.

Step 2: Apply Definition of Permutation

Permutation can be defined using factorial. The formula is given as \( P(n, r) = \frac{n!}{(n - r)!} \). Replace \(n\) and \(r\) with 7 and 4 respectively, we get \( P(7, 4) = \frac{7!}{(7 - 4)!} \).

Step 3: Evaluate Factorials

Factorial of a number is the product of all positive integers less than or equal to that number. Thus, we have \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \) and \((7 - 4)! = 3! = 3 \times 2 \times 1 = 6 \).

Step 4: Substitute Factorials into Formula

Substitute the values of \(7!\) and \(3!\) into the permutation formula, yields \( P(7, 4) = \frac{7!}{(7 - 4)!} = \frac{5040}{6}\).

Step 5: Simplify to Find Final Answer

Carrying out the division gives \( P(7, 4) = 840 \).

Key Concepts

Factorials

A factorial is a mathematical operation that multiplies a series of descending natural numbers. For any positive integer \( n \), the factorial is given as \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \). This operation is fundamental in permutations and combinations.

• The factorial of 0 is defined as 1, i.e., \( 0! = 1 \), even though there are no descending numbers to multiply.
• Factorials grow very fast. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Understanding factorials aids in comprehending how permutations and combinations work because they express the total number of ways to arrange items. This is crucial in solving problems involving permutations, such as \( _{7} P_{4} \).

Permutation formula

Permutations focus on arranging items, where the order matters. The formula for permutations is an essential tool:\[ P(n, r) = \frac{n!}{(n - r)!} \]Here, \( n \) represents the total items available, while \( r \) is the number of items to arrange. The formula gives the number of ways \( r \) items can be ordered out of \( n \) possible items. To break it down:

• \( n! \) calculates all possible arrangements of \( n \) items.
• \( (n-r)! \) divides by the redundant arrangements not involving selected \( r \) items.

This formula is employed to find permutations like \( _{7} P_{4} \), where we seek the arrangement of 4 items out of a total of 7.

Algebraic evaluation

Algebraic evaluation is a systematic approach to solving expressions by substituting known values. By substituting \( n = 7 \) and \( r = 4 \) into the permutation formula, we arrive at \( _{7} P_{4} = \frac{7!}{(7 - 4)!} \). Here’s how to tackle it:1. **Evaluate Factorials**: Calculate \( 7! = 5040 \) and \( 3! = 6 \). This stems from multiplying all numbers down to 1.2. **Substitution**: Plug these values into the formula.3. **Simplification**: Perform the division \( \frac{5040}{6} \) to find the solution, which results in 840.
This method makes solving permutation problems manageable by breaking them into smaller, logical steps. Algebraic evaluation helps clarify each part, leading to the final answer.