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Combinations

Arrangements & combinations

Japanese school year: Elementary 6

What you learn

Learn how to determine the total number of combinations when choosing items where the order of selection does not matter. This technique is practical for organizing tournaments, team selections, and everyday choices. Reviewing how to count permutations and structure organized tables will support your learning.

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Key points

This formula finds the number of combinations when choosing rr items from nn distinct items without regard to order. It divides the permutations nPr{}_n\mathrm{P}_r by r!r! to remove duplicate orderings.

nCr=nPrr!=n!r!(n−r)!{}_n\mathrm{C}_r = \frac{{}_n\mathrm{P}_r}{r!} = \frac{n!}{r!(n - r)!}

This formula calculates the number of ways to choose 2 items from nn items. It is commonly used to form pairs from nn people, count round-robin matches, or count lines connecting points.

nC2=n(n−1)2{}_n\mathrm{C}_2 = \frac{n(n - 1)}{2}

This identity shows that choosing rr items is equivalent to choosing which (n−r)(n - r) items to leave behind. For example, selecting 8 items from 10 is much easier to compute as 10C2{}_{10}\mathrm{C}_2.

nCr=nCn−r{}_n\mathrm{C}_r = {}_n\mathrm{C}_{n - r}

This identity states that there is exactly 1 way to choose 0 items (choosing none) and 1 way to choose all nn items from nn items. It serves as a foundational rule in combinations.

nC0=nCn=1{}_n\mathrm{C}_0 = {}_n\mathrm{C}_n = 1

Choose a set to practice.