I am sure I know the answer to this question but it is very important that it is correct. Can someone please double check for me to make sure there are no mistakes?

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QUESTION: If three (fair) dice are rolled what is the probability of rollingat mosttwo matching numbers? what is the probability of rollingat leasttwo matching numbers?

ANSWER: total possible outcomes of rolling 3 dice = 6*6*6= 216

two matching pair can be achieved by rolling

[x, x, 1]

[x, x, 2]

[x, x, 3]

[x, x, 4]

[x, x, 5]

[x, x, 6]

where x = any number from 1-6

Notice that at x=1 the first set is [1,1,1] which is more than two matching numbers. likewise at x=2 the second set would be [2, 2, 2] and so on for each number, x. therefore there are 5 possibilities in this case for each number in the set.

six numbers each with five possible outcomes = 30 possible outcomes in this set

Also two matching numbers could be achieved by

[x, 1, x]

[x, 2, x]

[x, 3, x]

[x, 4, x]

[x, 5, x]

[x, 6, x]

or

[1, x, x]

[2, x, x]

[3, x, x]

[4, x, x]

[5, x, x]

[6, x, x]

again each number, x has 5 possible outcomes giving us an additional 30 outcomes for each of these two sets.

30*3 = 90

divide by total possible outcomes

90/216 = .4167

AT LEAST two matching numbers includes the possiblity of rolling triples which can occur in six ways [1, 1, 1], [2, 2, 2], [3, 3, 3] and so on

for a total probability of 6/216

adding this to the previous answer we get 6/216 + 90/216 =96/216 = .4444

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again it is extremely important that these answers and the explanations are 100% correct. if anyone could double check the solution for me it would be greatly appreciated.

Thank you