Problem 19 - Algebra for College Students

Question

In how many ways can the 37 seats on a commuter flight be filled from the 39 people holding tickets?

Step-by-Step Solution

Answer

741 ways

Step 1: Understand the Problem

The problem is to determine the number of ways to fill 37 seats on a commuter flight with 39 ticket holders. This is a combinatorial problem where we must choose 37 people out of 39.

Step 2: Identify the Combinatorial Formula

We use the combinations formula to find the number of ways to select 37 people from 39. The combination formula is given by \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, and \( k \) is the number of items to choose.

Step 3: Apply the Formula

Here, \( n = 39 \) and \( k = 37 \). Plugging these values into the combination formula: \[ C(39, 37) = \frac{39!}{37!(39-37)!} = \frac{39!}{37! \times 2!} \]

Step 4: Simplify the Expression

Simplify the factorial expression: \[ \frac{39!}{37! \times 2!} = \frac{39 \times 38 \times 37!}{37! \times 2!} = \frac{39 \times 38}{2!} \]

Step 5: Compute the Final Answer

Calculate the denominator \( 2! = 2 \), so the expression becomes: \[ \frac{39 \times 38}{2} = \frac{1482}{2} = 741 \] Thus, there are 741 ways to fill the 37 seats from the 39 people holding tickets.

Key Concepts

Combinations

To solve the problem of finding out how many ways we can fill 37 seats from 39 ticket holders, we use the concept of combinations. Combinations count the number of ways to choose a subset of items from a larger set, where the order does not matter. The formula for combinations is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Here, \( n \) is the total number of items (39 ticket holders), and \( k \) is the number of items to choose (37 seats). The symbol \( ! \) denotes a factorial, which we'll discuss shortly. Using our values, the formula becomes: \[ C(39, 37) = \frac{39!}{37!(39 - 37)!} = \frac{39!}{37! \times 2!} \] This simplifies to give us: \[ \frac{39 \times 38}{2!} = \frac{1482}{2} = 741 \] So there are 741 ways to choose 37 people out of 39, meaning there are 741 ways to fill the seats on the flight.

Factorials

To understand the combination formula better, we need to know what factorials are. A factorial, denoted by \( n! \), is the product of all positive integers up to \( n \). For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

In the combination formula \[ C(n, k) = \frac{n!}{k!(n-k)!} \] the factorials help us calculate the different ways to choose \( k \) items out of \( n \) without worrying about the order. For our specific problem, we computed: \[ \frac{39!}{37! \times 2!} \] This essentially means we are calculating the product of numbers up to 39, then dividing by the product of numbers up to 37 and the product of numbers up to 2. Factorials grow very fast, so they're often used in combinatorial calculations.

Permutations

While this problem used combinations, another important concept in combinatorics is permutations. Permutations, unlike combinations, account for the order of items. For example:

• The combination of selecting letters A, B, and C is the same no matter the order (ABC, ACB, BAC, BCA, CAB, CBA all count as one).
• In permutations, all different orders are counted as unique.

The formula for permutations is: \[ P(n, k) = \frac{n!}{(n-k)!} \] Here, \( n \) is the total number of items to pick from, and \( k \) is the number of items to pick. If our problem were to find the number of ways to arrange 37 seats out of 39 in a specific order, we would use permutations instead of combinations. Understanding both combinations and permutations is crucial for solving various counting problems in mathematics and real-life applications, like seating arrangements, passwords, and more.