Understanding Combinatorics in Algebra

Combinatorics is the study of counting, arrangements, and combinations. It is essential in probability, algebra, and computer science.

Permutations

Permutations are arrangements of objects where order matters.

Formula: \( n! \) for \( n \) objects.

Combinations

Combinations are selections of objects where order does not matter.

Formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

The Binomial Theorem

The binomial theorem describes the expansion of \( (a + b)^n \).

Practice Problems

1. How many ways can you arrange 4 letters?
2. How many ways can you choose 3 out of 7 objects?
3. Expand \( (x + y)^3 \) using the binomial theorem.