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:
- Calculate 15! (factorial of 15).
- Then, calculate (15 - 9)! = 6!.
- 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:
- Let n = 12 (total lunches) and r = 4 (lunches to arrange).
- 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:
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