Statistics/Probability/Combinatorics

Combinatorics studies permutations and combinations of objects chosen from a sample space. A preliminary knowledge of combinatorics is necessary for a good command of statistics.

Counting Principle
The Counting Principle is similar to the Multiplicative Principle. If a process involves $$n$$ steps and the $$i$$th step can be done in $$x_i$$ ways, then the entire process can be completed in $$x_1 \times x_2 \times ... \times x_n$$ different ways.

Permutations
A permutation is a distinct arrangement of $$n$$ elements of a set. By the Counting Principle, the number of possible arrangements of $$n$$ objects in a set is $$n \times (n-1) \times (n-2) \times ... \times 1 = n!$$. What if some of the elements are not distinct? Then, if there are $$k$$ distinct kinds of elements, the total number of possible arrangements is $$\frac{n!}{n_1! \times n_2! \times ... \times n_k!}$$. What if we are arranging the elements in a circle, rather than a line? Then the number of permutations is $$(n-1)!$$.

When faced with very large factorials, a useful approximation is Stirling's formula: $$n! \approx \sqrt{2 \pi n} (\frac{n}{e})^n$$

Now suppose we only choose r distinct elements from the set (without replacement). Then the number of possible permutations becomes $$\frac{n!}{(n-r)!} = _nP_r$$.

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 $$\frac{_nP_r}{r!} = \frac{n!}{r! \times (n-r)!} = _nC_r = \displaystyle {n \choose r}$$.

Note also that $$\sum_{r=0}^k \displaystyle{m \choose r} \times \displaystyle{n \choose k-r} = \displaystyle{m+n \choose k}$$

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

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

$$(x + y)^2 = x^2 + 2xy + y^2$$

$$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$$

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

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

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

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