Problem 37 - Basic Statistics for Business and Economics with Student

Question

An overnight express company must include five cities on its route. How many different routes are possible, assuming that it does not matter in which order the cities are included in the routing?

Step-by-Step Solution

Answer

There is 1 possible route.

Step 1: Understand the Problem

We need to find out how many different ways five cities can be chosen for a delivery route where order does not matter. This is a combination problem.

Step 2: Recall the Combination Formula

The problem is a combination problem because the order of the cities does not matter. The formula to calculate combinations is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Apply the Formula

In this scenario, we have 5 cities, and we need to select all of them in any order. Thus, \( n = 5 \) and \( r = 5 \). Plug these into the formula: \[ \binom{5}{5} = \frac{5!}{5!(5-5)!} = \frac{5!}{5!\cdot 0!} \]. Since \( 0! = 1 \), the expression simplifies to \( \frac{5!}{5!} = 1 \).

Step 4: Interpret the Result

The calculation shows that there is only 1 way to choose the five cities, which means there's only 1 possible route when order does not matter.

Key Concepts

Combination Formula

When dealing with problems where the order of selection does not matter, the combination formula comes into play. This is key in combinatorics, as opposed to permutations where order does matter. The combination formula can be expressed in algebraic terms as:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \( n \) denotes the total number of items to choose from, while \( r \) refers to how many of those items are selected at a time. The "!" symbol indicates a factorial, which is the product of all positive integers up to a given number. The formula efficiently reduces the count of selections by accounting for duplicate sequences (i.e., order). For example, choosing 5 cities out of 5 has a factorial of zero in the denominator, signifying that there's just one choice when duplicates are irrelevant.

Permutation and Combination

Understanding the distinction between permutations and combinations is essential in combinatorics. Permutations are used when the order of items matters; combinations ignore order.

• **Permutations**: Imagine assigning awards to different students; first place differs from second.
• **Combinations**: Picture forming committees where the selection order of members does not change the team.

The selection of routes in an overnight express company's scenario is a combination problem. This is due to the fact that the order in which cities are visited does not change the overall route. Therefore, the number of distinct routes only depends on the number of cities involved, further exemplified by calculating a single combination \( n = r \), yielding exactly one possible route as computed.

Mathematical Problem Solving

Mathematical problem solving involves breaking down a problem into manageable steps and utilizing the right formulas or methods to find the solution. Here, we tackle the problem as follows:

• **Understand the Scenario**: Identify whether the problem requires permutations or combinations. Here, order doesn’t matter, marking it as a combination task.
• **Apply the Correct Formula**: Use the combination formula \( \binom{n}{r} \), recognizing \( n = r = 5 \) in our example.
• **Compute**: Simplify the factorials carefully, acknowledging special factorial cases like \( 0! = 1 \).
• **Interpret Results**: Conclude logically from computations, ensuring the answer's relevance and sense in context.

By approaching problems methodically, one can efficiently tackle daunting questions and arrive at solutions that are both coherent and logical. This strategic method fosters better understanding and retention of mathematical concepts.