Question

If 4 dice are rolled together, what is the number of ways in which at least 1 die shows 3? and If 4 dice are rolled together, what is the number of ways in which all 4 dice show 3

Please help immediately! give me a proper equation with it too on how to figure it out thanks!!

Answer 1

At least one three `=> 864`. All threes `=>1`

Explanation:

We have 4 standard 6-sided dice. What are the number of ways to have at least one three rolled? And what are the number of ways to have all 4 dice show a 3?

Let's first say that the dice are Blue, Red, Orange, and Yellow.

The second question is the more straight-forward - there is only 1 way for all 4 dice to show a 3 - the B, R, O, and Y dice must all be 3. We can write it this way:

`1^4=1`

Now we look at the question of having at least 1 die show a 3. To do that, let's work through this bit by bit.

First, let's set 1 die at 3, which means the other 3 dice can be any number at all (from 1 to 6, that is...). So when the B die is a 3, the other three dice can be any value, and so there are 6 options for each die. Further, we can set the B die to be 3, or the R die, or the O die, or the Y die, meaning we need to multiply by 4. This gives:

`4xx6^3=4xx216=864`