A fair $6$-sided die is rolled 5 times and the result is recorded for each roll. How many different results are possible? Of the possible results, in how many ways can there be a result containing exactly $2$ rolls of a $4$?
For the first part I got $6^5 = 7776$ different results are possible.
For the second part I am lost at what steps I should take. I was thinking $(\frac16)^2$ for rolling a $4$ twice, but the result doesn't make sense to me of $\frac1{36}$??
Any guidance would be greatly appreciated.
Best Answer
You know, from the first part, that there are $7776$ total possibilities. How many of these have exactly two fours?
Imagine the rolls ordered from $1$ to $5$. We can choose which rolls the fours will be rolled on as a combination, since there are five possible slots and two fours to place:
$$\binom{5}{2} = \frac{5 \cdot 4}{2}= 10$$
After we've chosen the fours, we have three slots remaining and five possibilities for each slot (since there can't be any more fours). Our final answer is $$10 \cdot 5^3 = \boxed{1250}$$