10.5 Permutations and Combinations

Permutations An arrangement of objects in which order is important Example 1a: Find the number of permutations of the letters in the word TIGER. 5 options 4 options 3 options 2 options 1 option

How to express this as a formula? Let’s investigate! Example 1b: In how many ways can you arrange 3 of the letters of the word TIGER? How to express this as a formula? Let’s investigate! 5 options 4 options 3 options Formula for permutations of n objects taken r at a time

Circular Permutations Five people are seated around a table: In a row (linear permutation): A B C E D ABCDE B A C D E BCDEA C A B D E CDEAB D A B C E DEABC E A B C D EABCD In a circle (circular permutation): There is only one circular permutation, but there are five corresponding linear ones. For a set with n members,

Example 2 How many circular permutations are possible when seating five people around a table? =24 permutations

Combinations An arrangement of objects in which order is NOT important Example 2: Count the possible combinations of 2 letters chosen from the list A,B,C & D? AB AC AD BA BC BD CA CB CD DA DB DC List permutations Remove repeats There are 6 possible combinations.

Example 3: Count the possible combinations of 3 letters (out of 5) chosen from the word TIGER? In Example 1b, we counted 60 different ways to arrange 3 letters chosen from the word TIGER when order is important. Now, since order is NOT important, divide 60 by the number of ways to arrange the 3 letters that were chosen. Number of arrangements for the 3 positions: So, Formula for combinations:

Example 4: If there are 9 students in a class, how many ways can a group of 1 to 3 students be formed? Three separate choices for the number of students: = 9 ways 1 student: 9C1= = 36 ways 2 students: 9C2= = 84 ways 3 students: 9C3= Total: 84+36+9 = 129 ways

Probability ORDER IS NOT IMPORTANT Example 5: From a standard deck of 52 cards, 5 cards are dealt. What is the probability that you receive only face cards? ORDER IS NOT IMPORTANT

= = = = = Expansions of for small values of n Pascal’s Triangle 1 1 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

From Pascal’s Triangle: Investigate

Investigate =

Binomial Theorem Example 6: Use Binomial Theorem to write the expansion of

Example 7: What is the coefficient of the term in the expansion of ? The term involves So we want

Calculating Permutations and Combinations on TI-84 Enter value for n (e.g. [5]) For permutations, press [MATH], highlight PRB and select [2:nPr] Enter value for r (e.g. [2]) Press [ENTER] For combinations, follow same steps except select [3:nCr]

Calculating Permutations and Combinations on TI-89 For permutations Press [2nd][MATH][7][2] Enter value for n (e.g. [5]) Press [,] Enter value for r (e.g. [2]) Press [ENTER] For combinations, follow same steps except press [2nd][MATH][7][3]