Algebra 1 Glencoe McGraw-Hill JoAnn Evans Math 8H 12-3 Permutations and Combinations
An arrangement or listing in which order is important is called a permutation. Examples: the placement of the top four finishers in a school spelling bee the 5 starters on a basketball team the five numbers in a student I.D. number Placing the finishers means arranging them in order, 1 st, 2 nd, 3 rd, 4 th. Each player has a specific position to play. An I.D. number of is not the same as 51743; the order makes an I.D.number unique.
The number of permutations of n objects taken r at a time can be found with this formula: The number of objects The number taken at a time Order matters in a permutation! Order matters in a permutation! Order matters in a permutation! Order matters in a permutation! Order matters in a permutation!
Evaluate the permutation: To save some time, you could have stopped expanding the numerator at 3! and cancelled common factors.
Since the order of the contestants is important, this situation is a permutation of 11 contestants taken 2 at a time. Definition of permutation There are 11 contestants in a state quiz bowl competition. The top two contestants will go on to compete at a national quiz bowl competition. How many different ways can 1 st and 2 nd place be chosen? 1st2nd
Since the order of the digits in a code is important, this situation is a permutation of 10 digits taken 6 at a time. A bike lock requires a six-digit code made up of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. No number can be used more than once. How many different codes are possible?
Since the order of the members chosen to be officers is important, this situation is a permutation of 14 club members taken 4 at a time. A club with 14 members wants to choose a president, vice-president, secretary, and treasurer. How many different sets of officers are possible?
Order definitely matters in a code. This is a permutation of 7 digits taken 7 at a time. A computer program requires a user to enter a 7-digit registration code made up of the digits 1, 2, 4, 5, 6, 7, and 9. Each number has to be used, and no number can be used more than once. How many registration codes are possible? 0! is equal to one. Strange, but true.
An arrangement or listing in which order is NOT important is called a combination. Examples: a club with ten members wants to choose a committee of four members choosing 3 out of 9 possible pizza toppings choosing 12 pencils from a box of 36 pencils The committee has no specific jobs or order; it’s just four members. The order of the toppings doesn’t make a difference in the pizza. The set of pencils in not based on any particular order or arrangement.
The number of combinations of n objects taken r at a time can be found with this formula: The number of objects The number taken at a time Order doesn’t matter in a combination!
The order of the books is not important. She just wants to choose 6 out of 12 titles, so this is a combination, not a permutation. Mrs. Pooley needs to choose 6 books out of 12 for her 7 th grade English class to read next year. How many different ways can she choose 6 books?
This problem is different from the first one about the club. Here, the order of the members chosen to go is not important, it’s just a group of 4 members. This is a combination of 14 club members taken 4 at a time. A club with 14 members wants to choose 4 members to attend a book club convention. How many different groups of 4 are possible?
A math teacher needs to choose 10 students out of 15 to serve as peer tutors for struggling algebra students. How many different groups of 10 are possible?