How many possible outcomes will there be to my Minecraft lock?

I just built a lock for a door in Minecraft. The user has to pick the correct four colors out of twelve. Order doesn't matter. What is the probability of the user guessing the correct colors?

Solution 1:

1. You have ⁴/₁₂ of a chance of picking the first colour right.

2. You have ³/₁₁ of a chance of picking the second colour right from the remaining eleven colours, if you got the first right.

3. You have ²/₁₀ of a chance of picking the third colour right from the remaining ten colours, if you got the former two colours right.

4. You have ¹/₉ of a chance of picking the fourth colour right from the remaining nine colours, if you got the former three colours right.

Hence, the probability of guessing the combination is: ⁴/₁₂ × ³/₁₁ × ²/₁₀ × ¹/₉ = 495^-1.

In other words, there are 495 different combinations, of which one is correct.

Obviously, all of this is useless, unless you are working on a exploration mode map. It's much faster to just take a pick and destroy your door.

Solution 2:

Use the Choose function. You have 12 items, from which you can choose 4 and the order of those chosen 4 does not matter. This means you want to calculate 12 choose 4.

n choose k = n! / (k! * (n - k)!)

So, you have 12!/(4!8!) = 495 different choices, giving you a probability of choosing the one correct answer 1/495 = .00202... = 0.2%

Maths.