Pencil Grouping Question

Question

There are 20 red pencils, 13 green, 12 yellow and 15 blue. If they are evenly distributed into groups of 3 with different colors, how many groups will you have? This is not a combinations/permutations question, just a straight up how many groups of 3 can you create where the pencils in each group is a different color.

I had to do this by hand but I am wondering if there is a formula for this type of thing. Below is what I did, resulting in 20 groups. I used red all the way through since there were more red pencils than any other color. Then I kept the others colors balanced, rotating between them. Is there a more efficient way to tackle this, like a formula?



start with 20 red pencils, 13 green, 12 yellow and 15 blue

group #1 rgb

leaving 19 red pencils, 12 green, 12 yellow and 14 blue

group #2 rgb

leaving 18 red pencils, 11 green, 12 yellow and 13 blue

group #3 ryb

leaving 17 red pencils, 11 green, 11 yellow and 12 blue

group #4 rgb

leaving 16 red pencils, 10 green, 11 yellow and 11 blue

group #5 ryb

leaving 15 red pencils, 10 green, 10 yellow and 10 blue

group #6 rgy

leaving 14 red pencils, 9 green, 9 yellow and 10 blue

group #7 rgb

leaving 13 red pencils, 8 green, 9 yellow and 9 blue

group #8 ryb

leaving 12 red pencils, 8 green, 8 yellow and 8 blue

......... keep doing this.for a while.....

1 red pencils, 0 green, 1 yellow and 1 blue

group #20 ryb

leaving 0 red pencils, 0 green, 0 yellow and 0 blue

Answer 1

Add them all up

20 + 13 + 12 + 15 = 60

Divide by 3 = 20

If one color has more than this (like if red has 21), then there would be no place to put pencils of that color that would be in a group where there isn't already one of those colors.

So just compare the number of the color with the largest amount in it to the number of groups that COULD be created (total divided by 3) and as long as it isn't more, the answer would be the total possible number of groups (total divided by 3)