Combination Formula with a lockout twist

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    Combination Formula
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SUMMARY

The discussion focuses on calculating combinations with a "lockout" constraint, specifically for selecting 6 items from 18 choices divided into 6 groups of 3. The formula C(n,r) is initially referenced, where n=18 and r=6, but the unique twist requires that once an item is selected from a group, the other items in that group are unavailable for selection. The problem suggests starting with smaller combinations to understand the setup better, such as (1,3) and (2,6), before tackling the full problem.

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  • Understanding of combinatorial mathematics, specifically combination formulas.
  • Familiarity with the concept of "lockout" constraints in selection problems.
  • Basic knowledge of group theory as it applies to combinations.
  • Ability to visualize and represent combinations in a structured format.
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Combination Formula with a "lockout" twist

Hi! I am trying to figure out all possible combonations for 6 items among a group of 18 choices. So I turn to my old friend C(n,r) to calculate where n=18 and r=6. "But WAIT!" I tell you before you hastily begin scribbling, "There is a twist..." You see my problem is that the items are divided up into 6 groups, with 3 choices in each group. Once a choice has been made in a group for the combination the other 2 in the group are unavailable, or "locked out" of the rest of the combination. The order doesn't necessarily matter but a choice must be selected from each of the six groups. Here's a visual representation:

A B C
1 A1 B1 C1
2 A2 B2 C2
3 A3 B3 C3
4 A4 B4 C4
5 A5 B5 C5
6 A6 B6 C6

If "B1" is selected in a single combination then "A1" and "C1" cannot be apart of the same combination. What is the formula for this and how many possible combinations are there?
 
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Try to start with a smaller problem (1,3) and (2,6) which you can count and then calculate this one. Honestly, I didn't quite understand the setup.
 

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