My understanding is that some problems can be solved both in ordered way and unordered way while some are more difficult and you have to choose one.
There are two problems that seem very similiar to me but one uses a permutation while other uses a combination. I would like to know why.
1. Classic birthday problem. As you know, $23$ people is what it takes to have the probability of a same bithday to be greater than $50\%$. It is calculated this way.
Answer: $1 - \frac{\textrm{permut}(365, 23) }{ 365^{23}}$.
2. Coin flip problem. Flip a coin 100 times. $P(\textrm{50 heads})$?
Answer: $\frac{\textrm{combin}(100, 50) }{ 2^{100}}$.
Why is permutation the correct formula for the first problem while combination is correct for the second problem?
Here's another problem. This one uses a combination with a multinomial coefficient.
3. Roll 6 dice. $P(\textrm{3 pair})$? Example of a 3 pair would be, $(1, 1), (2, 2), (3, 3)$.
Answer: $\frac{\textrm{[combin}(6, 3) \cdot 6!/(2!2!2!)]}{ 6^6}$
I think combining the combination with the multinomial coefficient, it orders the outcomes to some degree. Why do you have to order the outcomes in this problem while you don't in problem #2?
Thanks in advance for your help.