Problem 44 - Algebra and Trigonometry

Question

These problems involve distinguishable permutations. Jogging Routes A jogger jogs every morning to his health club, which is eight blocks east and five blocks north of his home. He always takes a route that is as short as possible, but he likes to vary it (see the figure). How many different routes can he take? [Hint: The route shown can be thought of as ENNEEENENEENE, where \(E\) is East and \(N\) is North.]

Step-by-Step Solution

Answer

He can take 1287 different routes.

Step 1: Understand the Problem

The jogger needs to travel 8 blocks east and 5 blocks north for each route. All paths that take the shortest route possible must consist of a combination of exactly 8 'E's and 5 'N's, representing east and north moves respectively.

Step 2: Calculate Total Steps

Each route consists of a total of 13 moves (8 to the East and 5 to the North). This can be thought of as arranging the letters in a sequence of 'EEEEEEEENNNNN'.

Step 3: Apply the Permutation Formula

To find the number of different sequences consisting of 8 'E's and 5 'N's, use the formula for permutations of a multiset: \[ \frac{13!}{8! \times 5!} \].

Step 4: Calculate Factorials

Calculate the factorials: 13! = 6,227,020,800; 8! = 40,320; 5! = 120.

Step 5: Compute the Permutations

Substitute the factorials into the permutation formula: \[ \frac{13!}{8! \times 5!} = \frac{6,227,020,800}{40,320 \times 120} = \frac{6,227,020,800}{4,838,400} = 1287 \].

Key Concepts

Factorials

Factorials are essential when dealing with permutations and combinations in mathematics. The factorial of a number, denoted as \(n!\), is the product of all positive integers up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials play a crucial role in calculating permutations. This is because they help count the total number of ways in which a set of items can be arranged.

• When we calculate \(13!\), we are finding the total number of ways to arrange 13 distinct items.
• Similarly, calculating \(8!\) and \(5!\) helps us understand the arrangements of respective subsets.

When you divide \(13!\) by \(8! \times 5!\), you are effectively accounting for repeated permutations of the subsets (where order does not matter within each subset). This results in only considering unique arrangements of the letters 'E' and 'N' in the problem of shortest jogging routes.

Multisets

A multiset is a generalized concept of a set that allows for multiple occurrences of its elements. In our jogging routes problem, the sequence of steps—consisting of both 'E' for east and 'N' for north—is an example of a multiset.

Unlike traditional sets, where each element can appear only once, multisets can have repeated elements. Utilizing a multiset for permutations involves accounting for identical items.

• For example, consider the sequence of 13 moves ('EEEEEEEENNNNN').
• This consists of 8 'E' moves and 5 'N' moves.

Permutations of a multiset are calculated using a specific formula to ensure no identical sequences are counted more than once. This is why we use the formula \( \frac{13!}{8! \times 5!} \), which divides the total arrangements of the multiset by factorials of repeated elements.

Combinatorial Counting

Combinatorial counting is a fundamental technique used to solve problems involving counting arrangements and selections of objects. It consists of different methods, such as permutations and combinations, that provide a structured approach to counting without having to list all possibilities.

In permutations, specifically permutations of multisets as in our exercise, we're interested in counting distinct arrangements of a sequence.

• The formula \( \frac{n!}{k_1! \times k_2! \times \ldots \times k_m!} \) is applied where \( n \) is the total count of items and \( k_1, k_2, ..., k_m \) are the counts of indistinguishable items in the multiset.
• Such an approach reduces complexity by avoiding manual enumeration of each sequence.

This technique allows us to solve problems like the jogging route problem efficiently, determining that 1287 distinct routes are possible while traveling the shortest path.