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