In combinatorics
The coefficient of in the expansion of gives the number of different ways to draw cards from two identical sets of playing cards each.[4] For example, from two sets of the three cards A, B, C, the different drawings are:
More information Number of selected cards, Number of options ...
Number of selected cards |
Number of options |
Options |
0 |
1 |
|
1 |
3 |
A, B, C |
2 |
6 |
AA, AB, AC, BB, BC, CC |
3 |
7 |
AAB, AAC, ABB, ABC, ACC, BBC, BCC |
4 |
6 |
AABB, AABC, AACC, ABBC, ABCC, BBCC |
5 |
3 |
AABBC, AABCC, ABBCC |
6 |
1 |
AABBCC |
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For example,
- .
In particular, this provides the formula for the number of different hands in the card game Doppelkopf.
Alternatively, it is also possible to arrive at this expression by considering the number of ways of choosing pairs of identical cards from the two sets, which is the binomial coefficient . The remaining cards can then be chosen in ways,[4] which can be written in terms of the binomial coefficients as
- .
The example above corresponds to the three ways of selecting two cards without pairs of identical cards (AB, AC, BC) and the three ways of selecting a pair of identical cards (AA, BB, CC).