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Example 1:
A youth baseball team has 15 players. How many different ways are there for the team's manager to select and arrange 9 of these players to make up the team's batting order?

Example 2:
Your school cafeteria has purchased enough food for 12 different lunches over the next few weeks. Due to a holiday on Monday, there are only 4 school days this week. How many different ways are there for the cafeteria to select and arrange 4 of these lunches to serve?

Answer :

Final answer:

The question involves using permutations to count arrangements of selections. In the first example, there are 1,630,000 arrangements of baseball players, while the cafeteria can arrange lunches in 11,880 ways. Both examples illustrate fundamental counting techniques in combinatorics.


Explanation:

Understanding Arrangements in Combinatorics

The questions presented relate to the concept of combinations and permutations in combinatorics, which is a branch of mathematics focused on counting, arrangement, and combination of objects.

Example 1: Youth Baseball Team

In the first example, to determine how many different ways the manager can select and arrange 9 players out of 15, we use the permutations formula. The formula for permutations is given by: P(n, r) = n! / (n - r)!.

Here, n is the total number of players (15) and r is the number of players to arrange (9). Plugging in these values:

  1. Calculate 15! (factorial of 15).
  2. Then, calculate (15 - 9)! = 6!.
  3. Apply the formula: P(15, 9) = 15! / 6!

This evaluates to 1,630,000 different arrangements.

Example 2: School Cafeteria Lunches

In the second example concerning the school cafeteria, we want to find the number of ways to select and arrange 4 different lunches from a total of 12 available options. Similar to the first example, we will use the permutations formula:

  1. Let n = 12 (total lunches) and r = 4 (lunches to arrange).
  2. Using the permutations formula: P(12, 4) = 12! / (12 - 4)! = 12! / 8!

Solving this gives us 11,880 different arrangements of lunches.

In both examples, we see the application of the permutations formula as a fundamental counting principle in mathematics.


Learn more about permutations here:

https://brainly.com/question/23283166


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