A factorial is a product written n!, meaning n times every positive integer down to 1. In Intermediate Algebra, you use it most often when counting arrangements and finding binomial coefficients in the Binomial Theorem.
A factorial is the product of a whole number and every positive whole number below it, written with an exclamation mark: n! = n × (n - 1) × (n - 2) × ... × 2 × 1. In Intermediate Algebra, factorials show up when you are counting how many ways something can be arranged or when you are working with combinations and binomial coefficients.
A small example helps. 5! means 5 × 4 × 3 × 2 × 1, which equals 120. That is much larger than 5 because factorials grow fast. Even one extra number changes the result a lot, which is why factorials are so useful in counting problems.
One special case is 0! = 1. That can feel odd at first, but it keeps the counting formulas consistent. For example, in combination formulas, you want expressions to work smoothly even when you choose all or none of the items.
Factorials are not used like ordinary multiplication inside regular equations. They are usually part of a formula, especially n! in permutations, combinations, and the Binomial Theorem. The biggest mistake is treating n! like n times 1 only, or forgetting to expand all the way down to 1. Another common error is leaving 0! as 0, when it is actually defined as 1.
In this course, factorials connect algebra to counting. If you see a question about rearranging objects, choosing items, or finding coefficients in an expansion, factorials are usually the tool that turns the problem into a formula you can compute.
Factorials matter in Intermediate Algebra because they are part of the formulas that count outcomes without listing every possibility. When you work with permutations, combinations, or the Binomial Theorem, the factorial notation tells you how many ordered or unordered selections are possible.
That makes factorials a bridge between algebra and counting. Instead of writing out every arrangement by hand, you can use n! to condense the pattern into one expression. This is especially helpful in problems where the numbers get bigger fast, because a direct listing would be messy or impossible.
Factorials also make binomial coefficients work. The formula n! / (k!(n-k)!) gives the coefficients in expansions like (x + y)^n, which connects to polynomial expansion and Pascal’s Triangle. If you know how factorials work, those coefficients become a calculation, not a mystery.
A lot of students first meet factorials in simple counting, then see them again inside larger formulas. Once you recognize the pattern, you can move faster on algebra problems that involve arrangements, combinations, and expansion.
Keep studying Intermediate Algebra Unit 12
Visual cheatsheet
view galleryPermutation
Permutations count arrangements where order matters, and factorials are built into the formula. If you are arranging 4 distinct objects, you are really using 4! to count the possible orders. That is why factorials show up whenever the problem asks how many different sequences or lineups are possible.
Combination
Combinations count selections where order does not matter, and the combination formula uses factorials in the denominator and numerator. The factorials help remove repeated orderings so you count each group only once. If you can tell whether order matters, you can usually decide whether to use a permutation or a combination.
Binomial Coefficient
Binomial coefficients are the numbers that appear in binomial expansions, and they are written with factorials as n! / (k!(n-k)!). Factorials are what make the coefficient formula work. In the Binomial Theorem, they tell you which term gets which numeric multiplier.
Polynomial Expansion
Polynomial expansion often uses factorial-based coefficients when you expand a binomial like (a + b)^n. The factorials are not the expansion itself, but they help generate the numbers in front of each term. That is why factorials matter when the course moves from simple multiplication to structured expansion patterns.
A quiz problem might ask you to evaluate a factorial, simplify a combination, or find a binomial coefficient. Your job is to expand the factorial correctly, cancel terms when a formula allows it, and keep track of whether order matters. For example, 6! is not 6 + 5 + 4 + 3 + 2 + 1, it is 6 × 5 × 4 × 3 × 2 × 1. In binomial expansion questions, factorials usually appear inside n! / (k!(n-k)!) and you may need to simplify before you calculate. If you miss the 0! = 1 rule, combination problems can fall apart, so that is a common check your teacher may expect you to know cold.
Factorial and permutation are related, but they are not the same thing. A factorial is a product notation, like 6!, while a permutation is a counting method for ordered arrangements. You often use factorials inside permutation formulas, which is why they get mixed up.
A factorial, written n!, means n times every positive integer down to 1.
Factorials grow very quickly, so they are useful for counting arrangements and combinations.
You need to remember that 0! = 1, not 0.
Factorials show up inside permutation, combination, and binomial coefficient formulas.
If order matters, factorials may be part of a permutation setup; if order does not matter, they often appear in a combination formula.
Factorial in Intermediate Algebra is the product of a whole number and all positive whole numbers below it, written n!. For example, 4! = 4 × 3 × 2 × 1 = 24. You see it in counting problems, combinations, and binomial expansion.
0! is defined as 1 so counting formulas stay consistent. This lets combination and binomial formulas work even when you choose all items or none. It is a definition, not a typo.
Expand it if you need the actual value, like 5! = 5 × 4 × 3 × 2 × 1. If you are simplifying a fraction with factorials, cancel matching factors first. That is common in combination and permutation problems.
A factorial is a notation for multiplying down to 1, while a permutation counts ordered arrangements. Permutation formulas usually contain factorials, but the terms are not interchangeable. If a problem asks about order, you are probably in permutation territory.