I have sets
$A = \{a_1, a_2, a_3\} $
$B = \{b_1, b_2\} $
$C = \{c_1, c_2, c_3\} $
and I'm trying to get combinations which are something like this:
$[a_1]$, $[a_1, b_1]$, $[a_1, b_1, c_1]$, $[a_1, b_1, c_2]$, $[a_1, b_1, c_3]$, $[a_1, b_2]$, $[a_1, b_2, c_1]$, $[a_1, b_2, c_2]$, $[a_1, b_2, c_3]$, $[a_1, c_1]$, $[a_1, c_2]$, $[a_1, c_3]$ and so on...
Am basically trying to get every possible combination (what is in the square brackets), where:
- It's ok if there is only one element in the square bracket
- Only one element from a set can be in a square bracket. ie: You can't have $[a_1, a_2]$
- Element pairs will never be found, reordered. ie: You'll never have
$[a1, b1]$ and $[b1, a1]$. It will always be $[a1, b1]$ only.
I know that if there are 2 items in one set and 3 items in another set, the number of ways they can be ordered are $2\cdot 3 = 6$ ways. And the $\frac{n!}{k!(n-1)!}$ rule of combinations.
But somehow, I find it hard to get to a formula for calculating the number of possible combinations for the sets I've depicted above.
ps: This isn't a homework question.