Collation – all possibilities

In the last collation post I covered calculating the odds of getting all four different Dugout Dirt cards from a box of 1994 Topps Stadium Club Series 1. Each box contains just four Dugout Dirt cards. The calculation gave the odds at 9.375%.

Another method to determine the odds is by looking at all the possibilities. There are 256 possible ways of pulling the four cards from the box. All 256 are shown below. Each possibility is shown as the order of the different cards. For example, 1-3-2-2 means that #1 was pulled first, then #3, then #2, and finally another #2.

1-1-1-1  1-1-1-2  1-1-1-3  1-1-1-4
1-1-2-1  1-1-2-2  1-1-2-3  1-1-2-4
1-1-3-1  1-1-3-2  1-1-3-3  1-1-3-4
1-1-4-1  1-1-4-2  1-1-4-3  1-1-4-4
1-2-1-1  1-2-1-2  1-2-1-3  1-2-1-4
1-2-2-1  1-2-2-2  1-2-2-3  1-2-2-4
1-2-3-1  1-2-3-2  1-2-3-3  1-2-3-4
1-2-4-1  1-2-4-2  1-2-4-3 1-2-4-4
1-3-1-1  1-3-1-2  1-3-1-3  1-3-1-4
1-3-2-1  1-3-2-2  1-3-2-3  1-3-2-4
1-3-3-1  1-3-3-2  1-3-3-3  1-3-3-4
1-3-4-1  1-3-4-2 1-3-4-3  1-3-4-4
1-4-1-1  1-4-1-2  1-4-1-3  1-4-1-4
1-4-2-1  1-4-2-2  1-4-2-3 1-4-2-4
1-4-3-1  1-4-3-2 1-4-3-3  1-4-3-4
1-4-4-1  1-4-4-2  1-4-4-3  1-4-4-4
2-1-1-1  2-1-1-2  2-1-1-3  2-1-1-4
2-1-2-1  2-1-2-2  2-1-2-3  2-1-2-4
2-1-3-1  2-1-3-2  2-1-3-3  2-1-3-4
2-1-4-1  2-1-4-2  2-1-4-3 2-1-4-4
2-2-1-1  2-2-1-2  2-2-1-3  2-2-1-4
2-2-2-1  2-2-2-2  2-2-2-3  2-2-2-4
2-2-3-1  2-2-3-2  2-2-3-3  2-2-3-4
2-2-4-1  2-2-4-2  2-2-4-3  2-2-4-4
2-3-1-1  2-3-1-2  2-3-1-3  2-3-1-4
2-3-2-1  2-3-2-2  2-3-2-3  2-3-2-4
2-3-3-1  2-3-3-2  2-3-3-3  2-3-3-4
2-3-4-1 2-3-4-2  2-3-4-3  2-3-4-4
2-4-1-1  2-4-1-2  2-4-1-3 2-4-1-4
2-4-2-1  2-4-2-2  2-4-2-3  2-4-2-4
2-4-3-1 2-4-3-2  2-4-3-3  2-4-3-4
2-4-4-1  2-4-4-2  2-4-4-3  2-4-4-4
3-1-1-1  3-1-1-2  3-1-1-3  3-1-1-4
3-1-2-1  3-1-2-2  3-1-2-3  3-1-2-4
3-1-3-1  3-1-3-2  3-1-3-3  3-1-3-4
3-1-4-1  3-1-4-2 3-1-4-3  3-1-4-4
3-2-1-1  3-2-1-2  3-2-1-3  3-2-1-4
3-2-2-1  3-2-2-2  3-2-2-3  3-2-2-4
3-2-3-1  3-2-3-2  3-2-3-3  3-2-3-4
3-2-4-1 3-2-4-2  3-2-4-3  3-2-4-4
3-3-1-1  3-3-1-2  3-3-1-3  3-3-1-4
3-3-2-1  3-3-2-2  3-3-2-3  3-3-2-4
3-3-3-1  3-3-3-2  3-3-3-3  3-3-3-4
3-3-4-1  3-3-4-2  3-3-4-3  3-3-4-4
3-4-1-1  3-4-1-2 3-4-1-3  3-4-1-4
3-4-2-1 3-4-2-2  3-4-2-3  3-4-2-4
3-4-3-1  3-4-3-2  3-4-3-3  3-4-3-4
3-4-4-1  3-4-4-2  3-4-4-3  3-4-4-4
4-1-1-1  4-1-1-2  4-1-1-3  4-1-1-4
4-1-2-1  4-1-2-2  4-1-2-3 4-1-2-4
4-1-3-1  4-1-3-2 4-1-3-3  4-1-3-4
4-1-4-1  4-1-4-2  4-1-4-3  4-1-4-4
4-2-1-1  4-2-1-2  4-2-1-3 4-2-1-4
4-2-2-1  4-2-2-2  4-2-2-3  4-2-2-4
4-2-3-1 4-2-3-2  4-2-3-3  4-2-3-4
4-2-4-1  4-2-4-2  4-2-4-3  4-2-4-4
4-3-1-1  4-3-1-2 4-3-1-3  4-3-1-4
4-3-2-1 4-3-2-2  4-3-2-3  4-3-2-4
4-3-3-1  4-3-3-2  4-3-3-3  4-3-3-4
4-3-4-1  4-3-4-2  4-3-4-3  4-3-4-4
4-4-1-1  4-4-1-2  4-4-1-3  4-4-1-4
4-4-2-1  4-4-2-2  4-4-2-3  4-4-2-4
4-4-3-1  4-4-3-2  4-4-3-3  4-4-3-4
4-4-4-1  4-4-4-2  4-4-4-3  4-4-4-4

Of the 256 possibilities, 24 (bolded) give rise to four different cards. That’s 24 out of 256, or 9.375%. That’s the same percentage that the calculation provided.

This method of determining odds works great, but it is a lot of work. Even just a small set (4 cards) gives a large number of possibilities. What would this look like for a 100 card set? This method, although it works, just isn’t going to be feasible for a normally sized set. Next week I’ll cover yet another option for determining the odds.

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