Question

A standard six-sided die is rolled $5$ times. You are told that among the rolls, there was one $4,$ one $5,$ and three $6$'s. How many possible sequences of rolls could there have been? (For example, $6,5,6,6,4$ is one possible sequence.)

Answer 1

Total possible sequences of rolls is 20.

Explanation:

A standard six-sided die is rolled $5$ times.

It is given that you are told that among the rolls, there was one $4,$ one $5,$ and three $6$'s.

We need to find possible sequences of rolls could there have been.

Total numbers to arrange (i.e.,4,5,6,6,6)= 5

Repleted numbers (i.e., 6,6,6) = 3

`5P_5=(5!)/((5-5)!)=5* 4* 3* 2* 1=120`

`3P_3=(3!)/((3-3)!)=3* 2* 1=6`

Total possible ways are

`Total=(5P_5)/(3P_3)`

`Total=(120)/(6)`

`Total=20`

Therefore, the total possible sequences of rolls is 20.

Answer 2

20

Explanation:

Data provided in the question:

Number of times a die is rolled = 5

Number repeating

4 - 1 times

5 - 1 times

6 - 3 times

Now,

The number of possible sequences of rolls can be the factorial of number of times the dice is rolled to the factorial of the number to times a number is repeated

Therefore,

The number of possible sequences of rolls =
`(5!)/(1!*1!*3!)`

or

The number of possible sequences of rolls =
`(5*4*3!)/(1*1*3!)`

or

The number of possible sequences of rolls = 20