Factorial

In mathematics, a factorial (denoted by an exclamation mark !) represents the product of all positive integers less than or equal to a given non-negative integer. For example, 5! (read as "five factorial") is calculated as 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, probability, and algorithm analysis, especially when dealing with permutations, combinations, and the calculation of possible arrangements of items. They grow rapidly, making them useful in scenarios with a large number of elements.

Factorial meaning with examples

• A group of 6 friends wants to line up for a photo. The number of different ways they can arrange themselves is calculated using the factorial of 6, (6!) which is 6*5*4*3*2*1 = 720. This demonstrates how factorials determine permutation possibilities, showcasing the various orderings.
• A lottery requires you to pick 6 numbers out of 49. The total number of possible combinations (ignoring order) is closely related to the factorial of 6 and the factorial of (49-6=43). Factorials show up here when calculating the combinations.
• In analyzing the time complexity of sorting algorithms like quicksort, factorials help in representing worst-case scenarios, and the computational cost is sometimes directly or indirectly impacted. Big O notation commonly uses factorials.
• If we want to scramble the alphabet, we use permutations based on the factorial of 26 (26!). This allows calculation of the massive number of unique possible arrangements of the 26 letters. The factorial quantifies the potential number of arrangements.