This is a problem related to combination math.
I have two sets of elements : $A = \big(a,b,…,z \big)$ and $B = (1,2,…,26)$.
Each element from set $A$ can be paired with an element of set $B$, leading to a pair. I want to make couple of unique pairs so that each pair can be selected multiple times but a couple can only be picked once:
{(a,1), (b,2)} is one valid couple,
{(a,2)(b,2)} is not (2 is used twice in the couple).
{(b,2)(a,1)} is not valid because identical to {(a,1), (b,2)}.
{(a,1),(b,3)} is valid (a,1) already picked but (b,3) not picked.
How many couple of pairs can I make for this dataset?
How many triplets of pairs (per ex.: {(a,2)(b,2),(c,3)}?
If dataset $A$ is of size $m$, and dataset $B$ of size $n$, the formula I get on the paper is $S = \sum_{i=0}^{n-1}m\Big(2.(n-1)-2i)\Big)$, but I experimentally I count $2.\binom{n}{2}^2$ combinations.
Event if the second formula is correct, I'm really interested in the general formula (triplets, quadruplets,… of pairs).
Best Answer
You can make $2\cdot\binom{26}2^2$ couples. Here is the reasoning:
Choose two element from A and two elements from B. That gives $\binom{26}2^2$. There are $2$ ways to arrange them. e.g. with $(a,b)$ and $(1,2)$, you can do $\{(a,1),(b,2)\}$ or $\{(a,2),(b,1)\}$.