Combinatorics and Binomial Theorem

Learning Objectives

Apply the fundamental counting principle.

Distinguish between permutations and combinations and use their respective formulas.

Expand a binomial using the Binomial Theorem or Pascal's Triangle.

The Mathematics of Counting

Combinatorics is the area of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

Fundamental Counting Principle

If there are m ways to do one thing, and n ways to do another, then there are m × n ways of doing both.

Example: If you have 3 shirts and 4 pairs of pants, you have 3 × 4 = 12 possible outfits.

Permutations and Combinations (Revisited)

Permutation (Order matters): The number of ways to arrange k items from a set of n.

nPk = n! / (n-k)!

Example: How many ways can Gold, Silver, and Bronze medals be awarded to 8 competitors?

8P3 = 8! / (8-3)! = 8! / 5! = 8 × 7 × 6 = 336.

Combination (Order doesn't matter): The number of ways to choose k items from a set of n.

nCk = n! / (k!(n-k)!)

Example: How many ways can a committee of 3 be chosen from 8 people?

8C3 = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56.

The Binomial Theorem

This theorem provides a formula for expanding a binomial raised to any power: (x + y)ⁿ.

The coefficients of the expansion can be found using Pascal's Triangle or the combination formula nCk.

Pascal's Triangle:

1 (n=0)

1 1 (n=1)

1 2 1 (n=2)

1 3 3 1 (n=3)

1 4 6 4 1 (n=4)

Expansion of (x + y)⁴:

The coefficients from Pascal's triangle are 1, 4, 6, 4, 1.

The powers of x start at n and decrease to 0.

The powers of y start at 0 and increase to n.

Result: 1x⁴y⁰ + 4x³y¹ + 6x²y² + 4x¹y³ + 1x⁰y⁴

= x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Key Terms

Combinatorics: An area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
Fundamental Counting Principle: A rule used to count the total number of possible outcomes in a situation. It states that if there are n ways of doing something, and m ways of doing another thing after that, then there are n×m ways to perform both of these actions.
Permutation: An arrangement of a set of objects in which the order of the objects is important.
Combination: A selection of items from a collection, such that the order of selection does not matter.
Binomial Theorem: A formula for finding any power of a binomial without multiplying at length.