Combinations

Japanese school year: Math A

What you learn

Learn how to use combination formulas to determine the number of ways to choose items when order does not matter. Combinations are essential for team selections, lottery odds, and binomial expansion coefficients. Familiarity with permutations and factorial calculations provides an important foundation for working with these selection techniques.

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Key points

This formula calculates the total number of combinations when choosing $r$ items from $n$ distinct items without regard to order. Use it when only group membership matters.

{}_n\mathrm{C}_r = \frac{n!}{r!(n-r)!}

This property shows that choosing $r$ items from $n$ is equivalent to choosing the remaining $n - r$ items to leave out. For example, ${}_{10}\mathrm{C}_8 = {}_{10}\mathrm{C}_2$ , which simplifies calculations when $r$ is large.

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

This formula splits choosing $r$ items from $n$ into two cases: including a specific item or excluding it. It represents the principle behind Pascal's triangle.

{}_n\mathrm{C}_r = {}_{n-1}\mathrm{C}_{r-1} + {}_{n-1}\mathrm{C}_r

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Practice 1Coming soon