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Discrete Mathematics is a branch of mathematics dealing with objects that can assume only distinct, separated values. It contrasts with continuous mathematics, which deals with objects that can vary smoothly.

Discrete mathematics includes topics such as:

• Graph theory
• Combinatorics
• Number theory
• Logic

In this exercise, we focus on combinatorics, which is about counting, arranging, and finding patterns in sets. Since we are choosing donuts, we need to count the different ways to select them, a classic problem within this field. Combinations with repetition, which allows us to select the same item multiple times, is a key method used here.

The binomial coefficient is a fundamental concept in combinatorics. It is used to count the number of ways to choose a subset of a certain size from a larger set without regard for the order of the items. The binomial coefficient is denoted as \( C(n, r) \), where \( n\) represents the total number of items to choose from, and \( r\) is the number of items to choose.

The formula for the binomial coefficient is: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]. Here, \( n! \) is the factorial of \( n \), which means the product of all positive integers up to \( n \).

For instance, in this problem we needed to calculate \( C(32, 12) \), which represents the number of ways to choose 12 donuts from 32 (21 varieties with repetition). This formula helps in solving the problem accurately by providing the exact number of combinations possible.

Combinatorial Analysis deals with counting, arranging, and grouping items. It is especially useful when dealing with problems where the arrangement or selection of objects is crucial.

In the given exercise, combinatorial analysis is essential because we need to determine how many possible ways there are to choose 12 donuts from 21 varieties, allowing for repetition. This specific scenario is addressed by combinations with repetition.

The formula \[ C(n + r - 1, r) \] is pivotal for such calculation:

• \( n \) is the types of items (21 varieties)
• \( r \) is the items to choose (12 donuts)

Substituting the values, we get \[ C(21+12-1, 12) = C(32, 12) \].

Therefore, combinatorial analysis provides the structured approach to get a precise solution by breaking down a complex selection process into understandable and calculable steps.