Pascal's Triangle is a triangular array of binomial coefficients. Rows in the triangle are denoted as n and columns are denoted as k, and the nth kth location of the triangle is denoted as the binomial coefficient (n k).  

Pascal's Triangle

The binomial coefficient is the coefficient of the N^n term in the binomial expansion of (x + 1)^n. Binomial Coefficients are widely used in combinatorics and probability. It is expressed by the following formula: 

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Given N people, the number of different ways to choose, for example, an n-person committee where the order doesn’t matter (i.e., everyone has the same interchangeable role) is denoted by the binomial coefficient and is the same formula as ordered sets repetitions not allowed, but includes division by the factorial of n to account for repeated permutations (see formula).

Unordered sets, repetitions not allowed

This is helpful when you want the binomial coefficient of a set of objects where the objects or unordered and repetitions are allowed.

Unordered sets, repetitions allowed

Morin, D. (2016). Probability for the Enthusiastic Beginner. Las Vegas, Nevada: CreateSpace Independent Publishing Platform.

Probability for the Enthusiastic Beginner by David Morin

Binomial Coefficient

A multiset, which is an unordered collection of elements where repetitions are allowed, can be represented using set-like notation. For example, a multiset containing multiple instances of the same element 'x' is denoted as `{x, x, ..., x}`.

Notation for a Multiset

A food truck offers 5 different types of sauces for its tacos. A customer wants to order 3 tacos and can choose any combination of sauces, including getting the same sauce on multiple tacos. Since the tacos are all placed in a single container, the order in which they are prepared does not matter. How many different combinations of sauces can the customer choose for their 3 tacos?

Based on the case study below, analyze the engineer's mistake. What specific, incorrect assumption about the selection process led to the wrong total? Explain why this assumption doesn't fit the scenario and describe the correct characteristic that should have been considered instead.

Inventory System Logic Error

A teacher wants to distribute 10 identical pieces of candy among 4 students. Any student can receive any number of candies, from zero up to all ten. Since the candies are identical, the specific combination of candies each student receives is what matters, not the order of distribution. How many distinct ways can the teacher distribute the candy?

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