Counting in Probability: A Dev's 2x2 Cheat Sheet

I enjoy probability and statistics, but counting problems often confuse me - especially with questions of replacement and order in experiments.

Here's a 2X2 matrix that handles these 4 possibilities:

Selection Type \ Order	Order Matters	Order Doesn't Matter
With Replacement	n^r	C(n+r-1,r)
Without Replacement	P(n,r) = n!/(n-r)!	C(n,r) = n!/(r!(n-r)!)

Where:

• n = total items to choose from
• r = items being chosen
• P(n,r) denotes Permutation
• C(n,r) denotes Combination

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To classify any counting problem, ask two questions:

• Can I use the same item again?
• Does sequence matter?

Examples:

1. Candy jar

   Allows replacement - can pick a candy, return it, and pick it again

2. Deck of cards

   No replacement - once drawn, a card stays out of the deck

3. Password

   Allows replacement - can reuse letters, and order matters

4. Committee

   No replacement - each person picked once, order doesn't matter