AI Tutor
Imagine expanding (x+y)40(x + y)^{40} and needing just the coefficient for x12y28x^{12}y^{28}. Building Pascal's triangle all the way down to Row 40 would take hundreds of additions across dozens of pages.
Fortunately, a single formula calculates any entry in any row instantly, powered by a mathematical operation called the factorial.

The Factorial Tool

A factorial, written with an exclamation mark as n!n!, is the product of all positive integers from nn down to 1. For example, 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120.
In 1808, French mathematician Christian Kramp introduced the exclamation mark notation simply because printers kept running out of custom symbols for large products. By definition, mathematicians also set zero factorial as 0!=10! = 1, representing the single way to arrange an empty set.
Factorials tell you how many ways you can arrange items in a line, but what happens when you only want to pick a smaller team where arrangement order doesn't matter?

The Combination Formula

A combination, written as nCr_n\text{C}_r or (nr)\binom{n}{r}, counts how many subsets of size rr you can choose from a total pool of nn distinct items when the internal order does not matter.
To count combinations, we take all n!n! orderings, divide out the (n−r)!(n - r)! items we leave behind, and divide out the r!r! duplicates caused by rearranging the chosen group.
Let's test this with a small concrete calculation: evaluating (52)\binom{5}{2}, which represents choosing 2 items from 5.
Writing out the definition gives (52)=5!2!(5−2)!=5×4×3×2×1(2×1)(3×2×1)\binom{5}{2} = \frac{5!}{2!(5 - 2)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)}. Notice how the tail ends cancel out completely, simplifying to 202=10\frac{20}{2} = 10.