Combinations
Combination problem are counting problems that seek to answer questions such as:
- In how many ways can 4 nursing students be selected from a group of 12 nursing students for evening duty?
- In how many ways can a car dealer select 15 cars from his inventory of 105 brand new automobiles to be placed in a local mall on a weekend?
- In how many way can a poker hand of 5 cards be selected from a standard deck of 52 cards?
- Given a group of 25 drivers, in how many ways can 10 of the drivers be assigned to drive a BMW, and 12 of the drivers assigned to drive a
Suburau and the remaining drivers assigned to drive a Ford?
- In how many ways can a committee of 5 math students, 4 English students and 8 history students be selected from a group of 12 math students, 10 English students and 5 history students?
In each of these examples the arrangement or order of objects is of no consequence.
- For the nursing students, once the four students are selected it makes no difference where a name appears on the evening work list.
- For the automobile dealer, once the 15 automobiles are selected we will assume that it makes no difference where they are placed in the mall.
- Once the poker hand is drawn, the order in which the cards appear in the players hand is of no consequence.
- For the drivers once the first 10 drivers are selected, it does not matter which one of the 10 goes into which BMW. The same can be said for the other drivers.
- For the books
- For the math students once 5 students have been selected, order does not matter because neither math student out ranks any other math student. The same can be said for the other students in the group.
Now for the combination formula.
N! is read factorial
Lets revisit the first question. In how many ways can 4 nursing students be selected from a group of 12 nursing students for evening duty?
The answer is:
See how to use the TI Calculator to work this problem.
Now lets work the poker hand problem. In how many way can 5 cards be selected from a standard deck of 52 cards?
The answer is:
The player to receive the first five cards is receiving one out of a possible two million five hundred ninety eight thousand nine hundred sixty.