Combinatorics

• Fundamental Counting Principle: If you must make a number of separate decisions, then multiply the number of ways to make each individual decision to find the number of ways to make all the decisions.
• For problems in which certain choices are restricted and/or affect other choices, choose the most restricted options first.
• The number of ways of putting n distinct objects in order, if there are no restrictions, is n!
• When you have repeated items, divide the "total factorial" by each "repeat factorial" to count the different arrangements. 
• To count possible groups, divide the total factorial by two factorials: one for the chosen group and one for those not chosen.
• If you are required to choose two or more sets of items from separate pools, count the arrangements separately - perhaps using a different anagram grid each time. Then multiply the number of possibilities for each step.
• If the problem has unusual constraints, try counting the arrangements without constraints first. Then subtract the forbidden arrangements.
  - for problems in which items or people must be next to each other, pretend that the items "stuck together" are actually one large item.
  - keep in mind that stuck together items can be XY and YX so multiply the result by two.
• A combination is a selection of items from a larger pool, the order of items does not matter.
• A permutation is also a selection of items from a larger pool, the order of items matters.
• If switching the elements in a chosen set creates a different set, it is a permutation. If not, it is a combination.