Rolling Dice 5 Times

Question

A fair 6-sided die is rolled 5 times and the result is recorded for each roll.

a) How many different sequences of results are possible?

b) Of the possible sequences of results, how many of them contain exactly 3 rolls of a 4?


I'm pretty sure part (a) is $6^5 = 7,776$.

Part (b) is really confusing me though. I think that I should be using $ _nC_r$ somehow, but I'm not sure what $n$ and $r$ should be.

Answer 1

For (a), you are correct.

For (b), first, try to solve a simpler question:

How many results contain $4$ in their first three rolls, and no $4$ after that?

Now, answer another similar question:

How many results have a $4$ in the last three rolls, and no other fours?

And another similar one:

How many results have a $4$ in positions $2,4$ and $5$, and no other fours?

---

Once you answer that question, think about

1. Are the sets of results in those three questions disjoint?
2. how many different questions of that type you can ask?
3. How many results are there, therefore, in total?

Answer 2

There are $\binom53$ ways to select $3$ rolls out of $5$ that are destined to be the rolls that give a $4$.

For a selected roll there is $1$ possibility (it gives a $4$).

For a not selected roll (there are $2$ of them) there are $5$ possibilities (it gives no $4$).

Draw conclusions.

So that gives $\binom531^35^2=\binom535^2=250$ possibilities.