Statistics - Combination

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. we might ask how many ways we can select 2 letters from that set.

Combination is defined and given by the following function −

Formula

${C(n,r) = \frac{n!}{r!(n-r)!}}$

Where −

• ${n}$ = the number of objects to choose from.

• ${r}$ = the number of objects selected.

Example

Problem Statement

How many different groups of 10 students can a teacher select from her classroom of 15 students?

Solution

Step 1 − Determine whether the question pertains to permutations or combinations. Since changing the order of the selected students would not create a new group, this is a combinations problem.

Step 2 − Determine n and r

n = 15 since the teacher is choosing from 15 students.

r = 10 since the teacher is selecting 10 students.

Step 3 − Apply the formula

${^{15}C_{10} = \frac{15!}{(15-10)!10!} \\[7pt] = \frac{15!}{5!10!} \\[7pt] = \frac{15(14)(13)(12)(11)(10!)}{5!10!} \\[7pt] = \frac{15(14)(13)(12)(11)}{5!} \\[7pt] = \frac{15(14)(13)(12)(11)}{5(4)(3)(2)(1)} \\[7pt] = \frac{(14)(13)(3)(11)}{(2)(1)} \\[7pt] = (7)(13)(3)(11) \\[7pt] = 3003}$