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Solve advanced combinatorics problems instantly. Determine exactly how many possible arrangements or selections exist within massive datasets. This free online tool allows you to calculate combinations and permutations quickly and accurately. No sign-up or installation required.
In mathematics, Combinatorics is the study of counting, arranging, and grouping items. While counting seems like a simple task, the numbers involved in combinatorics explode into massive, astronomical figures almost instantly.
For example, if you simply want to arrange a standard 52-card deck, the number of possible unique arrangements is 52! (52 factorial). That number is an 8 followed by 67 zeros. If you shuffled a deck of cards once every second since the universe was born, you wouldn't even be close to seeing every possible arrangement. Combinatorics allows us to calculate and harness these massive probabilities accurately.
The difference between a Combination and a Permutation is the most common point of confusion in statistics. The entire mathematical formula hinges on answering one simple question: Does the order of the items matter?
Think of a fruit salad. If you put apples, bananas, and grapes into a bowl, it is the exact same fruit salad as one made with grapes, apples, and bananas. The order you picked them doesn't change the final group. Used for selecting a committee of 3 people from a group of 10, or picking lottery numbers.
Think of a PIN code for your bank account. 1-2-3-4 is a completely different code than 4-3-2-1. Even though the exact same numbers are used, the specific sequence makes it a unique arrangement. Used for race results (1st, 2nd, 3rd place), passwords, and scheduling.
A fun fact about this field of mathematics is that the world-famous "combination lock" you used in high school is mathematically named incorrectly!
Because the order of the numbers on a padlock strictly matters (dialing 10-20-30 will open it, but dialing 30-20-10 will not), it is technically a Permutation Lock. A true "combination lock" would open no matter what sequence you dialed, as long as you entered the right three numbers.
Expert clarification on factorials, repetition, and lottery probabilities.