What is the probability of getting one pair and one trio when throwing five dice?
I tried to answer this question stating that the probability of getting $3$ equal numbers is $(\frac{1}{6})^3$ and the probability of getting $2$ equal numbers in the rest of the dice is $(\frac{1}{6})^2$, but you can have $6$ number options in the trio and $5$ number options in the pair, so I concluded that the probability is $\frac{6\times5}{6^5} \approx 0.3858\text{%}$.
I run a Monte Carlo Simulation in Python to check this answer, and I got a different answer, roughly $10$ times the probability just calculated. That is the origin of posting this question. Thanks!
How I did it is that I split it into finding number of ways to get a trio and a pair, and number of permutations for $5$ dice rolls(I'm assuming we are dealing with $6$-sided dice here).
Number of ways to get condition
There are $6$ numbers we can chose for a pair($[1,1],[2,2]$, so on).
There are $\frac{5*4}{2}=10$ ways to chose the $2$ dice that are going to be the same.
There are $5$ choices(the other three have to be the same as well but not the same as the pair value) to choose the other values for the other $3$ dice.
So the number of ways to get one pair and one trio is $6*10*5$.
Number of ways to roll $5$ dice
There are $6^5$ ways to roll $5$ dice.
So, the probability is: $\frac{6*10*5}{6^5}=\frac{25}{648}\approx 3.858\text{%}$
(I'm not entirely sure of my answer so if you catch any errors please tell me)
Edit: Thanks Gerry Myerson for catching my error