<h3>
Answer: 220</h3>
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Explanation:
Imagine we have 3 slots to fill: slot A, slot B, slot C.
There are 12 choices for slot A
There are 11 choices for slot B (whatever was picked cant be reselected)
There are 10 choices for slot C
There are 12*11*10 = 1320 permutations. If order mattered, then we would be done with the problem.
However, order does not matter as there is no need to rank or sort the various colors. All we care about is the group rather than the individual colors.
For any single group of 3, there are 3*2*1 = 6 ways to arrange the objects. This means that we divide by 6 to go from a permutation to a combination
1320/6 = 220
So there are 220 combinations. This tells us there are 220 different groups of colors where the order does not matter in any single particular group.
Eg: group {A,B,C} is the same as group {B, C, A}
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An alternative method is to use the combination formula.
We would plug in n = 12 and r = 3 to get,
nCr = (n!)/(r!*(n-r)!)
12C3 = (12!)/(3!*(12-3)!)
12C3 = (12!)/(3!*9!)
12C3 = (12*11*10*9!)/(3!*9!)
12C3 = (12*11*10)/(3!)
12C3 = (12*11*10)/6
12C3 = 1320/6
12C3 = 220
we get the same answer as before.
