91Ó°ÊÓ

When creating an exercise routine, the sequence of exercises matters significantly. Todd has 12 different exercises to choose from, but he can only perform 9 at a time. To find out how many different combinations of these 9 exercises Todd can organize, we need to consider *permutations* rather than combinations.

• If the order of selection is important, like in an exercise routine, we use permutations.
• If the order doesn't matter, we use combinations.

So, in Todd's case, since he believes the sequence affects performance, permutations are the right choice. This distinction is crucial in discrete mathematics, where we often have to evaluate similar situations and choose the right mathematical tool.

A permutation is an arrangement of objects in a specific order. The permutation formula helps us calculate the number of ways to arrange a subset of items from a larger set. The formula is: \[ P(n, r) = \frac{n!}{(n - r)!} \] Here, *n* is the total number of items to pick from, and *r* is the number of items to arrange. In Todd's case:

• *n = 12* because he has 12 exercises to choose from.
• *r = 9* because he will perform 9 exercises.

Plugging in these values, we get: \[ P(12, 9) = \frac{12!}{(12 - 9)!} \] This formula shows the importance of both selecting and arranging items, which is fundamental in studies of permutations.

Factorials are a key element in calculating permutations and combinations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For example: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] In Todd's exercise problem, we utilize factorials to compute the permutation. We calculate:
* *12!* (the factorial of 12), which represents the total arrangements of all 12 exercises. * *(12-9)!* (the factorial of the difference between 12 and 9), which adjusts for the subset we're creating. \[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600 \] \[ 3! = 3 \times 2 \times 1 = 6 \] Dividing these, we get: \[ \frac{12!}{3!} = \frac{479001600}{6} = 79833600 \] Thus, Todd can create 79,833,600 unique exercise routines.

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. It deals with topics such as logic, set theory, graph theory, and combinatorics. In Todd's exercise routine problem, discrete mathematics is applied through:

• *Permutations*: Arranging a subset of exercises where order is important.
• *Factorials*: Calculating ways to arrange items.

Discrete mathematics is crucial for problems like these because it provides the tools to handle finite sets of objects and ensures accurate results for various arrangements and combinations. Understanding these principles supports problem-solving in many real-world contexts, like optimizing schedules or planning activities.