What is the sample size when we toss a dice 5 times, the answer is 6^5 but I want to know why?

Question

What is the sample size when we toss a dice 5 times, the answer is 6^5 but I want to know why?

Answer 1

There are 6 numbers on the first di.

There are 6*6 permutations of  2 di: 11, 12, ... 66

There are 6*6*6 permutations of 3 di: 111, 112, ..., 666

There are 6*6*6*6 permutations of 4 di: 1111, 1112, ..., 6666

There are 6*6*6*6*6 = 6^5 permutations of 6 di.

Answer 2

The sample size is how many possibilities/options you can have.

When you roll a die one time, it can come up with a 1, 2, 3, 4, 5, or 6.  There are 6 possibilities that you could get.

That means that it has a sample size of 6^1, or simply 6.

If you roll it twice, you could get 1 to 6 on the first roll, and independently, you could get 1 to 6 on the second roll.  In other words, you have 6 options each time you roll the die.

That means it has a sample size of 6x6, or 6^2, or simply 36.

Spelled out, you can have the two dice come up with:

1 & 1 - 1

1 & 2 - 2

1 & 3 - 3

1 & 4 - 4

1 & 5 - 5

1 & 6 - 6

2 & 1 - 7

2 & 2 - 8

2 & 3 - 9

2 & 4 - 10

2 & 5 - 11

2 & 6 - 12

3 & 1 - 13

3 & 2 - 14

3 & 3 - 15

3 & 4 - 16

3 & 5 - 17

3 & 6 - 18

4 & 1 - 19

4 & 2 - 20

4 & 3 - 21

4 & 4 - 22

4 & 5 - 23

4 & 6 - 24

5 & 1 - 25

5 & 2 - 26

5 & 3 - 27

5 & 4 - 28

5 & 5 - 29

5 & 6 - 30

6 & 1 - 31

6 & 2 - 32

6 & 3 - 33

6 & 4 - 34

6 & 5 - 35

6 & 6 - 36

So, each time you roll the die, you would have 6 times more possibilities.  If you roll the die 4 times, the sample size is 6^4.  Finally, if you roll it 5 times, the sample size is 6^5.

Hopefully this makes sense to you now.