Combinations
The last method of counting we will use to determine the number of ways events can occur involves combinations.
A combination represents the number of ways \(r\) objects can be selected from a group of \(n\) total objects, without repetition, when the order that the objects are selected is not important. So we often say this is an unordered arrangement of objects.
The last method of counting we will use to determine the number of ways events can occur involves combinations.
A combination represents the number of ways \(r\) objects can be selected from a group of \(n\) total objects, without repetition, when the order that the objects are selected is not important. So we often say this is an unordered arrangement of objects.
The formula for combinations is: \(\large{_nC_r=\frac{n!}{\left[r!\left(n-r\right)!\right]}}; n\ge r\), where \(n\) is the total number of objects to choose from and \(r\) is the number of objects being selected.
Example 1: Evaluate \(\large{_6C_2}\).
\(\large{_6C_2=\frac{6!}{\left[2!\left(6-2\right)!\right]}=\frac{6!}{2!\cdot4!}=\frac{6\cdot5\cdot4!}{2\cdot1\cdot4!}=\frac{30}{2}=\normalsize15}\)
Similar to permutations, the calculator can perform these calculations as well. The operation is located in the same menu as the permutation formula.
Some key situations and words to look for to help you decipher if the problem involves combinations include:
- Choose
- Select
- Group
- Committee
- Student council
- Jury
- Hand of cards
- Hand shakes (hugs, high fives, fist bumps, etc.—which are combinations of \(2\) people)
- Any other situation where the order of selection won’t matter
Example 2: Illinois has several lottery games. Once such game is Powerball. To win the Powerball jackpot, you must correctly match \(5\) numbers from \(1\) to \(69\) and one number from \(1\) to \(26\). The order in which the numbers are selected does not matter. How many winning combinations exist? (Remember: “and” means multiply.)
\(\large{_{69}C_5\cdot_{26}C_1=\normalsize11,238,513\cdot26=292,201,338}\) (I am letting the calculator do the work!)
There are $\(292,201,338\) possible combinations.
Example 3: There are \(20\) men and \(10\) women in a jury pool. How many \(12\) person juries can be formed that are made up of at most \(2\) women?