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Combinations

A combination is a set: order does not matter. Count the ordered picks, notice that every set shows up k! times, and divide — that is all nCr is.

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Choose a team of 3 from 5 people, A to E. A team has no order: ABC and CBA are the same team. Start from what the last page counts — ordered picks. There are P(5, 3) = 5 × 4 × 3 = 60 of them, listed here in dictionary order.

C(5, 3) = \frac{P(5, 3)}{3!} = \frac{60}{6} = \mathbf{10}\qquad \frac{5!}{3!\,(5 - 3)!} = \frac{120}{6 \cdot 2} = 10

n (people)5k (chosen)3The grid shows k up to 3: k = 4 would need 24 orderings per row

The top fraction is the picture: 60 chips in rows of 6. The bottom one is the same number written with factorials — k! removes the order inside the set, (n − k)! the order of the ones left out.

Combinations

A combination is a set: order does not matter. Count the ordered picks, notice that every set shows up k! times, and divide — that is all nCr is.

Step 1 of 17

Speed

C(5, 3) = \frac{P(5, 3)}{3!} = \frac{60}{6} = \mathbf{10}\qquad \frac{5!}{3!\,(5 - 3)!} = \frac{120}{6 \cdot 2} = 10

n (people)5k (chosen)3The grid shows k up to 3: k = 4 would need 24 orderings per row

The top fraction is the picture: 60 chips in rows of 6. The bottom one is the same number written with factorials — k! removes the order inside the set, (n − k)! the order of the ones left out.

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Combinations

See also

Combinations

See also