I have 4 different color dice: blue, red, yellow and green. I need to check how many possibilities we get if the set of the numbers that are on the dices consists of 3 distinct numbers. In other words, 1 dice is equal to another and the other two are different.
I have 2 answers in my mind and I'm not sure which is right:
- first : 6 * 5 * 4 * 3 (the last one is the 1 of the three options on the 3 dices we chose) so its 360.
- second : 6 * 5 * 4 * 6 (lets say we choose the first, 6 options, then the next one is the same so 1 option, after its going to be 5 options and than 4 options, and we do this 6 times...) so i get 720. exactly double.
I am not sure which one is right and why. Thank you in advance.
Step one: pick which two colored dice match. $\binom{4}{2} = 6$ possibilities
Step two: pick what number is on the matching dice. $6$ possibilities
Step three: pick the number of the unmatched die with the color earliest in lexigraphical order (I.e., if the green and yellow dice are the matching pair, I will choose the number for the blue die in step three, but if blue green are the matching pair I will choose the number for the red die in step three). $5$ possibilities (since we could not choose the same number as what we chose for the matching pair in step 2)
Step four: pick the number of the final unmatched die. $4$ possibilities (since we could not choose the number in step three or the number used in step 2)
For a final total of $6\cdot 6\cdot 5\cdot 4$ possibilities.