Combination numbers are one of the most used concepts in advanced math contests. The question is what exactly are combination numbers? To understand this, we must first look at a factorial.

What is a factorial?

A factorial is the product all integers from one to the factorial number. For example, 8 factorial is just 1×2x3×4x5×6x7×8 or 1×2x3…x8. The symbol for a factorial is !. 8 factorial can be shown as 8!.

How combination numbers work

Now that you understand what a factorial is, I will now explain how combination numbers work. Combination numbers are usually written like this: C(n,k). N is a number and K is a number. Basically, when a number is written in this form, it is just a simplification of N!/(K!(N-K)!). Now, let us plug in some numbers to see how this works. Let us say N=5 and K=3. The equation would be 5!/(3!(2!)). Remember that 5! and 3! both contain 1×2x3. These numbers will then cancel out giving you the result 5×4/2! = 5×4/(2)= 5×2=10. Let’s look at another problem, let us say n=8 and k=4. The problem would be set up as 8!/(4!(4!)). 8! and 4! both contain 1×2x3×4, giving you the answer, 8×7x6×5/4! = 8×7x6×5/(4×3x2×1). 3×2=6 so the 6s cancel out and you get 8×7x5/4. The 8 and the 4 can be simplified and you get your answer 2×7x5=70.

What combination numbers are used for

Combination numbers are used for a variety of different math problems. A problem with combination numbers might be if there are 8 people and 5 different spots places they can sit, how many distinct ways can you sit people and stand the rest? 8 would equal your N value and 5 would equal you K value. Your combination would therefore be 8!/(5!(3!). 5! and 8! simplify. 8×7x6/3!. 3! = 6, the 6s cancel out, you are left with your answer 8×7. There are 56 ways to seat people. If you have, any questions just ask me in the comments.