Counting / Restricted Arrangements
Least You Need to Know: Restricted Arrangements
Harder counting problems often become easier once you decide whether order matters, whether items repeat, and whether a restriction should be handled first or by subtraction.
The least you need to know
- If order matters, think permutations or arrangement stages.
- If order does not matter, think combinations.
- Restrictions often suggest fixing a block, choosing forbidden positions, or subtracting bad cases.
- Repeated items require a different count than distinct items.
- Read phrases like 'at least', 'together', 'not adjacent', and 'exactly' very carefully.
Key notation
nPr
ordered selections
nCr
unordered selections
!
factorial
C(n,r)
binomial coefficient
Tiny worked example
- Arrange the letters of BOOK.\n- There are 4 letters but two O's are identical.\n- Count = 4!/2! = 12.
Common mistakes
- Students often multiply when the problem really asks for a selection.
- Students often forget to divide by repeats for repeated letters.
- Students often ignore the restriction until the very end, when it is harder to fix.
How to recognize this kind of problem
- If the same chosen objects can appear in different orders, order matters.
- If identical items appear, ask whether different swaps really create new outcomes.
- For adjacency restrictions, consider a block method or count-total minus bad.