Permutations & Combinations

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Factorial

• The product of all positive integers to some number

Permutations

• An ordered arrangement of a set or items
• Used when order matters and elements are not-reused (unless specified)

Complete Permutation

• , where is the number of items
• Used when arranging all items in a set
• “How many ways can the letters in ‘Lizard’ be rearranged?”
  • “Lizard” → 6 characters;

Permutation with Replacement

• , where is the number of items in the set and is the number of items selected
  • 10 possible digits, 4 slots → 10,000 possible pin codes.
• Choosing items where order matters and repetitions are allowed

Permutation without Replacement

• Choose a subset of items where the order matters and elements cannot be duplicated

Permutation with Constraints

• “How many ways can you arrange the letters ‘abcdefg’ where ‘bcd’ must stay together”

Combinations

• Choosing items from a set of items where order does not matter

Combination without Replacements

• “How many 5-card hands in a 52-card deck?”

Combination with Replacements

• “Roll 3 5-sided dice, order is irrelevant”

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