\(P[\text{pair and 3 odd dice rolled}]=\dfrac{\dbinom{5}{2}\dbinom{6}{1}\dbinom{5}{3}3!}{6^5}=\dfrac{25}{54}\)

The factors in the numerator correspond to

a) pick 2 dice out of the 5 to be the pair

b) pick 1 value out of 6 to be the pair's value

c) pick 3 values out of 5 for the other dice

d) permute the 3 odd values since we do care about order here since we are scaling by the entire number of possible rolls

Now we roll the three odd dice to obtain a three of a kind of any value

There are only 6 ways of doing this, 1 per value.

\(P[\text{roll 3 of a kind}]=\dfrac{6}{6^3}=\dfrac{1}{36}\)

we multiply these together to get the final probability

\(P[\text{full house or 5 of a kind}]=\dfrac{25}{55}\dfrac{1}{36}=\dfrac{25}{1944}\)

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