Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ _{10} P_{4} \div 4 ! $$

Step-by-Step Solution

Answer

The result of the expression is 210.

Step 1: Understand the Symbols

The expression is \( _{10}P_{4} \div 4! \). Here, \( _{10}P_{4} \) refers to a permutation, which represents the number of ways to arrange 4 items out of 10. The symbol \( 4! \) (read as "4 factorial") means the product of all positive integers up to 4: \( 4 \times 3 \times 2 \times 1 \).

Step 2: Calculate the Permutation \( _{10}P_{4} \)

The formula for permutation is \( _{n}P_{r} = \frac{n!}{(n-r)!} \). For \( _{10}P_{4} \), this means: \[ _{10}P_{4} = \frac{10!}{(10-4)!} = \frac{10!}{6!} \]. Calculating further, we get: \[ 10 \times 9 \times 8 \times 7 = 5040 \] since \( 6! \) cancels out the rest of the \( 10! \).

Step 3: Calculate \( 4! \)

The factorial of 4, \( 4! \), is calculated by multiplying all integers from 1 to 4: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \].

Step 4: Divide the Permutation by 4!

Now, divide the result from the permutation by the factorial: \[ \frac{5040}{24} = 210 \].

Step 5: Conclusion

The final value after performing the division is 210.

Key Concepts

Understanding Factorials

Factorials are a crucial concept in permutations and combinations. The factorial of a non-negative integer \( n \) is denoted by \( n! \) and is defined as the product of all positive integers less than or equal to \( n \). For example, \( 4! \) means

• Multiply: \(4 \times 3 \times 2 \times 1 \)
• This equals: 24

Factorials are used to calculate the number of arrangements or sequences possible for a set of items.
They form the building blocks for both permutations and combinations. When \( n \) is 0 or 1, the factorial of \( n \) is defined as 1, which serves as the base case in computations.
Understanding how to compute and utilize factorials can significantly simplify solving permutation problems.

Combinatorics in Action

Combinatorics is the branch of mathematics that studies the counting, arrangement, and combination of objects. It is the foundation for understanding permutations and combinations.
When dealing with permutations, we are interested in the number of ways to arrange a set of items. This is different from combinations, which focus on the selection of items without considering the order.

• Permutations: Order matters. Calculate using factorials.
• Combinations: Order doesn’t matter. Typically fewer possibilities than permutations.

In permutations such as \( _{10}P_{4} \), combinatorics helps us determine how many ways we can arrange 4 items out of a total of 10. Both require a strong understanding of factorials, as they form the mathematical basis for combination and permutation calculations.

Exploring the Permutation Formula

The permutation formula is used to find the number of ways \( r \) objects can be arranged from a set of \( n \) objects. The formula is denoted as \( _{n}P_{r} \), which is computed by dividing the factorial of \( n \) by the factorial of \( n-r \):\[ _{n}P_{r} = \frac{n!}{(n-r)!} \]For example, to calculate \( _{10}P_{4} \):

• Compute \( 10! \), representing all possible arrangements of 10 items.
• Divide by \( 6! \) (i.e., \((10-4)!\)), which removes the arrangements not involving the 4 selected items.
• The result is: \(10 \times 9 \times 8 \times 7 = 5040\) arrangements.

This formula is essential for efficiently finding arrangements without manually listing them. It allows quick computation of permutations, making it easier to solve complex problems involving multiple items.