Problem 86 - Algebra and Trigonometry

Question

I used the permutations formula to determine the number of ways people can select their 9 favorite baseball players from a team of 25 players.

Step-by-Step Solution

Answer

The number of ways 9 players can be chosen from a team of 25 players can be calculated using the combination formula and the values for 'n' and 'r'.

Step 1: Identify 'n' and 'r'

In the given problem, n represents the total number of baseball players, which is 25, and r represents the number of players to be chosen, which is 9.

Step 2: Calculate Factorials

Calculate the factorials involved in the combination formula. So, calculate \(25!\), \(9!\) and \((25-9)!\).

Step 3: Apply Combination Formula

Substitute the calculated values into the combination formula to calculate the total number of ways to select 9 players out of 25. Deducing, \[ C(25, 9) = \frac{25!}{9!(25-9)!} \]

Step 4: Calculate Final Result

Perform the above calculations to find the final result.

Key Concepts

Factorials

Factorials are a fundamental concept in math, especially when dealing with permutations and combinations. They are denoted by an exclamation mark, and the factorial of a number, say 5, is written as \(5!\). A factorial is the product of all whole numbers from the specified number down to 1. For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 0! = 1 \) by convention

In combinations and permutations, factorials help us calculate the total arrangements or selections possible. The larger the number, the larger the factorial, increasing rapidly to very big numbers! This is why factorials are so crucial in working out these types of problems. Understanding factorials can ease the complexity involved when calculating with the formulas, as they are the building blocks of many calculations.

Combination Formula

When you are choosing items from a group where the order doesn't matter, you use combinations. This is exactly what was done with the baseball players—choosing favorites from a team. The combination formula is used to find out how many ways you can choose 'r' items from a group of 'n' items:\[C(n, r) = \frac{n!}{r! (n - r)!}\]In our baseball example, 'n' is 25 (total players) and 'r' is 9 (chosen players). The formula helps us determine the total possible combinations of choosing 9 players from 25, without considering the order.

• \( 25! \) represents all permutations of all players.
• \( 9! \) corrects for all arrangements of the chosen players.
• \( (25-9)! \) accounts for all possibilities of the remainders not chosen.

By substituting the values into the formula, it's straightforward to calculate and find the total number of possible combinations.

Permutations Formula

Permutations are used when the order of arrangement matters. However, in the particular baseball problem discussed, permutations aren't necessary since selection order isn't important. But generally, permutations allow us to calculate the different arrangements possible for a set of items where order is crucial.The permutations formula is:\[P(n, r) = \frac{n!}{(n - r)!}\]Here:

• \( n! \) is the factorial of the total number of items.
• \( (n-r)! \) is the factorial of the difference between the total and chosen items, removing the extra arrangements of remaining items.

This formula would give us the total number of ways to arrange 'r' items from a set of 'n' items.In the context of permutations versus combinations:- Use permutations if the order matters.- Use combinations if the order doesn't matter.It's crucial to recognize whether a problem asks for permutations or combinations to apply the correct formula. Understanding the difference can simplify otherwise complex problems.