Problem 21 - Algebra 2

Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. the winner and first, second, and third runners-up in a contest with 10 finalists

Step-by-Step Solution

Answer

This is a permutation problem with 10 finalists and 5040 possible orderings.

Step 1: Understanding the Difference between Permutations and Combinations

Permutations are used when the order of selection matters, whereas combinations are used when the order does not matter. In this problem, the order matters because the winner, first, second, and third runner-ups each have distinct positions.

Step 2: Breaking Down the Problem Requirements

We have 10 finalists, and we need to determine the order for 4 positions: winner, first runner-up, second runner-up, and third runner-up. This indicates a permutation because each position is unique and order matters.

Step 3: Applying the Permutation Formula

The formula for permutations of selecting r items from n items is given by \( P(n, r) = \frac{n!}{(n-r)!} \). Here, \( n = 10 \) and \( r = 4 \).

Step 4: Calculating 10 Factorial (10!)

Calculate \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \). This value will be used in the permutation formula.

Step 5: Calculating (10-4)! = 6!

Calculate \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). This is the denominator in the permutation formula.

Step 6: Substituting Values into the Permutation Formula

The number of permutations is \( P(10, 4) = \frac{10!}{6!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!} \). Simplifying gives \( 10 \times 9 \times 8 \times 7 \).

Step 7: Calculating the Final Answer

Calculate \( 10 \times 9 \times 8 \times 7 = 5040 \). Therefore, there are 5040 different ways to order the winner and the top three runner-ups.

Key Concepts

Combinations

In mathematics, combinations refer to the selection of items where the order does not matter. For example, if you are choosing 3 books from a shelf of 5, it does not matter in which order you pick them - this is a combination.

• Combination formula is used when the order of items is irrelevant.
• The formula for combinations is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \).

The important thing to remember is that combinations are about choosing items without reflecting their sequence. They are often used in scenarios such as forming committees, teams, or groups where the arrangement of selection doesn't impact the outcome.

Factorial

Factorial, represented by an exclamation mark (\(!\)), is a mathematical operation that multiplies a series of descending natural numbers. For instance, \(10!\) means multiplying all whole numbers from 10 down to 1: \(10 \times 9 \times 8 \times \ldots \times 2 \times 1\).

• Factorials are fundamental in calculating permutations and combinations.
• \( n! \) means you multiply \( n \) by every positive integer less than \( n \).

Factorials grow incredibly fast with large numbers and are crucial in fields like statistics, algebra, and calculus. They help simplify expressions and solve problems that include permutations and combinations.

Permutation Formula

The permutation formula is used when the order of the selection of items is important. The question "How many ways can we arrange these objects?" is solved using permutations. The formula for permutations of selecting \( r \) items from a total of \( n \) is:\[ P(n, r) = \frac{n!}{(n-r)!} \]

• It differs from combinations by considering different orders as separate outcomes.
• Use this formula when sequence or rank really matters, like in races or rankings.

For instance, arranging medals for races or ordering tasks are common scenarios where permutations are applied. Applied correctly, it helps determine the exact number of sequences possible.

Order of Selection

The order of selection is a key concept that differentiates between permutations and combinations. When talking about order, it concerns whether the sequence in which we select items carries significance. This aspect is the primary factor in choosing between calculating a permutation or combination.

• If the sequence matters, use permutations.
• If order doesn't matter, choose combinations.

In contests or ranking scenarios, as with the problem about ordering winners and runner-ups, knowing the order is vital - thus, permutations are used. In contrast, if you were selecting team members without assigning roles, order would not be important, hence a combination approach would apply.