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Combinations

A combination is essentially a subset. It is like a permutation, except with no regard to order. Suppose we have a set of $n$ elements and take $r$ elements. The number of possible combinations is $\frac{_{n} P_{r}}{r!} = \frac{n!}{r! \times (n - r)!} =_{n} C_{r} = (\binom{n}{r})$ .

Note also that $\sum_{r = 0}^{k} (\binom{m}{r}) \times (\binom{n}{k - r}) = (\binom{m + n}{k})$

Combinations are found in binomial expansion. Consider the following binomial expansions:

$(x + y)^{1} = x + y$

$(x + y)^{2} = x^{2} + 2 x y + y^{2}$

$(x + y)^{3} = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$

$(x + y)^{4} = x^{4} + 4 x^{3} y + 6 x^{2} y^{1} + 4 x y^{2} + y^{3}$

As you may have noticed from the above, for any positive integer $n$ , $(x + y)^{n} = \sum_{i = 0}^{n} [(\binom{n}{r}) \times x^{n - r} \times y^{r}]$

Another observation from the above is known as Pascal's law. It states that $(\binom{n}{r}) = (\binom{n - 1}{r}) + (\binom{n - 1}{r - 1})$

This allows us to construct Pascal's triangle, which is useful for determining combinations:

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Combinations

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