Trying to grasp some basic counting problems, but I have a hard time developing some intuition.
For example, if you roll 5 dice, the chance of getting a pair is 5 choose 2. But why? I can't wrap my head around it. Is this a shorthand for some more elaborate calculation? There are 6^5 total possibilities. How does this factor in to this? How does the '5' relate to this, apart from the 5 dice of course. Why aren't the outcomes a factor? Are these 'hidden'?
On
The probability for rolling one pair (and three singles) is: the probability for selecting four from six values, of which one from these four is for a pair, and arranging these as the outcomes among the roll of five dice. By arranging we mean: select two from five dice for the pair, and a die for each single among the three that remain.
$$\left.\dbinom 64\dbinom 41\dbinom 5{2}3!\middle/6^5\right.~=~\dfrac{5!^2}{2!^2~3!~6^4}$$
Thus is where the numbers originate
1 pair -- (which means that we are excluding 2-pair, 3-off a kind, etc)
We will see 4 different numbers.
$6\cdot 5\cdot 4\cdot 3$
one of those numbers is paired up.
Now we need to think about shuffling the dice.
It helps if you think of the dice as being different colors. This way it is more obvious why a $(x,x,x,y,y)$ is different from a $(y,x,x,y,x)$ i.e. same numbers in different order.
There are ${5\choose 2}$ ways to "shuffle the dice."
$6\cdot 5\cdot 4\cdot 3\cdot{5\choose 2} = 3600$