Counting Theory Permutation and Combination Starter 6 0

Counting Theory (Permutation and Combination)

Starter 6. 0. 1 • Suppose you have 6 different textbooks in your backpack that you want to put on a bookshelf. How many ways can the 6 books be arranged on the shelf?

Objectives • Use organized lists and tree (branching) diagrams to list all possible outcomes of a trial. • Identify whether permutation or combination is appropriate to count the number of outcomes of a trial. • Use formulas or calculator commands to evaluate permutation and combination problems.

Counting Outcomes • There are three principle ways to count all the outcomes of a trial. 1. Draw a tree diagram • Often a simple multiplication is enough 2. Systematically write all possibilities 3. Use permutation and combination techniques

Example: Tossing Coins • Three coins are tossed (or one coin is tossed three times) and the outcome of heads or tails is observed. • Draw a tree diagram (also called a branching diagram) that shows all possible outcomes. • State a conclusion: How many equally likely outcomes are there in this problem?

Tree Diagram for 3 Coins First Toss Second Toss Third Toss T H H H T T T H H T H T • So there are 8 different equally likely outcomes.

Write an Organized List • For the coin-toss problem we just did, write an organized list that shows all possible outcomes (like HHH etc) • Here is one possible organization – HHH, HHT, HTH, HTT – TTT, THT, TTH, THH • There are other ways to organize – Any method that is systematic (so that no outcomes are missed) can work

Permutations of n objects • Return to the bookshelf question. • Suppose we change the problem to arranging 10 books on the shelf. Now how many arrangements are there? • 10 x 9 x 8 x … = 3, 628, 800 – The shorthand notation for this is 10! (factorial) • In general, there are n! ways to arrange n objects – This is called the permutation of n objects – The key idea is that order matters

Arranging fewer than all the objects • What if there are only 4 slots available on the bookshelf for the 10 books? • Then there are 10 x 9 x 8 x 7 = 5040 ways to arrange 4 books out of a group of 10 • Notice that this could be viewed as • If we let n = the total number of objects and r = the number chosen and arranged, then we could conclude that the number of ways to arrange n objects taken r at a time is • This can be quickly evaluated by (n. Pr) – Try it now on your calculator with n = 10 and r = 4

Example • How many three-letter “words” can be made from the letters A, B, C, and D? – You can use your calculator to answer this. – What are n and r in this problem? • Don’t worry that many of them are not real words; we don’t care in this context. • Write an organized list of all the possible “words” – Be systematic; be sure you write them all.

Three-letter “words” ABC ABD ACD BCD ACB ADC BDC BCA BDA DCA CDB BAC BAD DAC CBD CAB DAB CAD DBC CBA DBA CDA DCB • Notice that being organized helps find all 24 permutations • Notice also that ABC is different from ACB because in permutation order matters • Suppose we don’t care about order. Then we are looking at combination, not permutation. • How many combinations of three letters can be made from an alphabet with four letters?

Three-letter “words” ABC ABD ACD BCD ACB ADC BDC BCA BDA DCA CDB BAC BAD DAC CBD CAB DAB CAD DBC CBA DBA CDA DCB • When order does not matter, ABC is the same as ACB (etc. ), so there are only 4 combinations in the 24 permutations. – They can be seen in the 4 columns • Notice that there are 3! (which is r!) permutations of each combination. – They can be seen in the 6 rows • So to get the number of combinations of n objects taken r at a time, divide permutations by r! • The formula is • The calculator command is (n. Cr) Try it now.

Examples • How many ways are there to form a 3 member subcommittee from a group of 12 people? • How many ways are there to choose a president, vice-president, and secretary from a group of 12 people?

More Examples • There are 5 cabins in the woods at a certain vacation spot. Each cabin has a path that leads to each of the other cabins. How many paths are there in all? – This is the combination of 5 things taken 2 at a time (order does not matter), so 5 C 2=10 • There are 100 communications satellites orbiting earth. Each satellite needs a transmit and receive channel to talk to each of the other satellites. How many channels are needed? – This time AB is different from BA, so use permutation: 100 P 2=9900 • How many pentagons can be drawn from the vertices of a regular 13 -gon? – Combination: 13 C 5=1287

Objectives • Use organized lists and tree (branching) diagrams to list all possible outcomes of a trial. • Identify whether permutation or combination is appropriate to count the number of outcomes of a trial. • Use formulas or calculator commands to evaluate permutation and combination problems.

Homework • Create and solve two story problems which illustrate the differences between combinations and permutations. – Create and solve a problem involving the permutation of n things taken r at a time. – Create and solve a problem involving the combination of n things taken r at a time.