Question

In how many ways can you select five people from a group of eight if the order of selection is important?

Short answer

Answer: There are 6,720 ways to select five people from a group of eight if the order of selection is important.

Step-by-step solution

Determine the number of objects (n) and the number of objects being selected (r)

In this problem, the total number of objects (n) is the eight people in the group, and the number of objects being selected (r) is five people.

Write the permutation formula

The permutation formula is P(n, r) = n! / (n - r)!. In this case, we are calculating P(8, 5).

Calculate (n - r)

Subtract r from n to get 8 - 5 = 3.

Calculate n! and (n - r)!

Compute the factorials for n and (n - r): 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 3! = 3 × 2 × 1 = 6

Divide n! by (n - r)!

Divide the factorial of n by the factorial of (n - r) to get the number of permutations: P(8, 5) = 40,320 ÷ 6 = 6,720

Interpret the result

There are 6,720 ways to select five people from a group of eight if the order of selection is important.

Key concepts

Factorial

Factorials are an essential mathematical concept used in permutations and combinations. The factorial of a non-negative integer, expressed as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials are used to calculate permutations where order matters, and combinations where order does not matter. In the calculation of permutations, we use the factorial of the total number of objects as well as the difference between total objects and objects selected. By understanding how factorials work, you can easily determine how items can be arranged or selected in various mathematical problems.

Combinatorics

Combinatorics is a field of mathematics that deals with counting, arranging, and finding patterns. It encompasses several important concepts like permutations and combinations. In permutations, the order of selection matters, while in combinations, it does not. Combinatorics helps solve problems related to selecting and arranging objects such as people, books, or other items.
Permutations fall under combinatorics as they deal with arranging a set number of items in specific orders. If you're selecting items and the sequence is vital, like the order of people in a line-up, permutations are used. Understanding combinatorics can greatly enhance problem-solving skills when it comes to calculating possibilities in various scenarios, especially when each scenario requires a distinct approach.

Order of Selection

The order of selection is crucial in determining whether to use permutations or combinations. If the order in which items are selected matters, we use permutations. For example, choosing which five people from a group of eight to perform distinct tasks requires tracking the specific order.
In permutations, changing the order creates a different arrangement, significantly impacting the total number of possible outcomes. On the other hand, if the order does not matter, then the problem deals with combinations, where order is irrelevant.

• Permutations: The order of selection is important, leading to more arrangement possibilities.
• Combinations: The order of selection is not important, focusing solely on the selection itself without regard to sequence.

Grasping this difference is critical for accurately solving problems and knowing when to apply each concept.