Combinatorics

Combinatorics is the branch of math that counts how many ways things can happen. It matters because many real problems involve arranging objects, choosing groups, or tracking multi step decisions. You see it in probability, computer science, genetics, scheduling, and game strategy. Learning the main counting tools helps you solve these problems efficiently instead of listing every case by hand.

The core idea is to match each situation to the right strategy. If order matters, you often use permutations, and if order does not matter, you often use combinations. For multi step processes, the multiplication principle and counting trees help organize outcomes clearly. Factorials, restrictions, and repeated objects add important variations, so careful interpretation of the problem is just as important as the formula.

Key Facts

Multiplication principle: if one step has a choices and the next has b choices, total outcomes = a x b.
Factorial: n! = n(n - 1)(n - 2)...(2)(1), with 0! = 1.
Permutations of r objects chosen from n distinct objects: nPr = n! / (n - r)!
Combinations of r objects chosen from n distinct objects: nCr = n! / [r!(n - r)!]
Order matters for permutations, but order does not matter for combinations.
Permutations with repeated objects: total arrangements = n! / (a!b!c!...) where a, b, c are counts of identical items.

Vocabulary

Factorial: A factorial is the product of all positive integers from 1 up to a given number, written with an exclamation mark.
Permutation: A permutation is an arrangement of objects where different orders count as different outcomes.
Combination: A combination is a selection of objects where different orders count as the same outcome.
Multiplication Principle: The multiplication principle says to multiply the number of choices at each independent step to get the total number of outcomes.
Counting Tree: A counting tree is a branching diagram that shows all possible choices step by step.

Common Mistakes to Avoid

Using combinations when order matters, which is wrong because arrangements like ABC and BAC are different outcomes in permutation problems.
Using permutations when order does not matter, which is wrong because it overcounts the same group multiple times in different orders.
Forgetting to reduce choices after an object is used without replacement, which is wrong because the number of available options changes from one step to the next.
Ignoring repeated identical objects, which is wrong because swapping identical items does not create a new arrangement and leads to overcounting.

Practice Questions

1 A school offers 4 math electives, 3 science electives, and 2 art electives. If a student chooses one course from each category, how many different schedules are possible?
2 How many different 3 letter arrangements can be made from the letters A, B, C, D, and E if no letter is repeated?
3 A class needs to choose a 4 person committee from 10 students. Explain why this is a combinations problem rather than a permutations problem.