Occupancy Probability Eight Sided Die
(Octahedron)

For this problem, we'll calculate the probabilty of getting all 8 numbers after rolling an 8-sided die 20 times.

STEP 1
The set of numbers on the die (1 through 8), would be called called "n" and the number of trials or attempts is called "r" which in this case is 20 rolls.
Calculating the value of "n" raised to the power of "r":

820
which equals
1,152,921,504,606,850,000

This number represents all the possible results from rolling an 8 sided die

In steps 2, 3, 4 and 5, we will determine how many of those 820 rolls, will contain all 8 numbers.

STEP 2
We first must calculate each value of "n" raised to the power of "r".
Rather than explain, this is much easier to show:

820 = 1,152,921,504,606,850,000
720 = 79,792,266,297,612,000
620 = 3,656,158,440,062,980
520 = 95,367,431,640,625
420 = 1,099,511,627,776
320 = 3,486,784,401
220 = 1,048,576
120 = 1

STEP 3
Next, we calculate how many combinations can be made from "n" objects for each value of "n".
This is much easier to show than explain:

8 C 8 = 1
7 C 8 = 8
6 C 8 = 28
5 C 8 = 56
4 C 8 = 70
3 C 8 = 56
2 C 8 = 28
1 C 8 = 8

Basically, this is saying that
8 objects can be chosen from a set of 8 in 1 way
7 objects can be chosen from a set of 8 in 8 ways
6 objects can be chosen from a set of 8 in 28 ways
5 objects can be chosen from a set of 8 in 56 ways
4 objects can be chosen from a set of 8 in 70 ways
3 objects can be chosen from a set of 8 in 56 ways
2 objects can be chosen from a set of 8 in 28 ways
1 object can be chosen from a set of 8 in 8 ways

STEP 4
We then calculate the product of the first number of STEP 2 times the first number of STEP 3 and do so throughout all 8 numbers.

1,152,921,504,606,850,000   ×   1     = 1,152,921,504,606,850,000
79,792,266,297,612,000   ×   8     = 638,338,130,380,896,000
3,656,158,440,062,980   ×   28     = 102,372,436,321,763,000
95,367,431,640,625   ×   56     = 5,340,576,171,875,000
1,099,511,627,776   ×   70   =   76,965,813,944,320
3,486,784,401   ×   56   = 195,259,926,456
1,048,576   × 28   = 29,360,128
1     ×   8     = 8

STEP 5
Then, alternating from plus to minus, we sum the 8 terms we just calculated.

 + 1,152,921,504,606,850,000 - 638,338,130,380,896,000 + 102,372,436,321,763,000 - 5,340,576,171,875,000 + 76,965,813,944,320 - 195,259,926,456 + 29,360,128 -8 Total   611,692,004,959,217,000

which equals the total number of ways you can roll an 8 sided die 20 times and have all 8 numbers appear.

STEP 6
So, if we take the number 611,692,004,959,217,000 and divide it by
1,152,921,504,606,850,000 (all posiible results of 20 rolls of an 8 sided die)
we get the probability of having all 8 numbers appearing after 20 rolls.

611,692,004,959,217,000 ÷ 1,152,921,504,606,850,000 = 0.530558240535038

So, if you had an 8 sided die, you would need to roll it at least 20 times in order to have a better than 50 / 50 chance of rolling all 8 numbers.