P&S - Definition of 'Combinations'

Combinations

${}_n{C}_r$ : number of combinations $n$ : total number of objects in the set $r$ : number of choosing objects from the set

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~ Ex.:

→ Source

It can be surprisingly difficult to count the number of sequences or sets satisfying certain conditions. For example, consider a bag of marbles in which each marble is a different color. If we draw marbles one at a time from the bag without replacement

How many different unordered sets (combinations)?

• 1 Marble ($r$ : number of choosing objects from the set)
• 1 Color ($n$ : total number of objects in the set) → ${}_n{C}_r$ : number of combinations = 1

1-2 Marbles, 2 Colors:

1-3 Marbles, 3 Colors:

1-4 Marbles, 4 Colors:

• 1 Marble, 4 Colours → 4 Combinations
• 2 Marble, 4 Colours → 6 Combinations
• 1 Marble, 4 Colours → 4 Combinations
• 1 Marble, 4 Colours → 1 Combinations