Lesson 42 Permutations and combinations factorial A factorial

Lesson 42 Permutations and combinations

factorial A factorial of a positive integer is the product of all positive integers up to and including that integer. 0 factorial (0!)is defined to be 1 n factorial the factorial of n is denoted as n! n! = n(n-1)(n-2)(n-3)…. . 1

Evaluating expressions containing factorials 5!= 5 x 4 x 3 x 2 x 1= 120 (7 -1)!= 6 x 5 x 4 x 3 x 2 x 1= 720 10! = 10 x 9 x 8!= 90 = 45 2!(10 -2)! 2 x 1 (8!) 2 8! = 8 x 7 x 6 x 5 x 4 x 3!=1680 2! 2! 3! 2 x 1 x 3!

practice 5! 2!(5 -2)! + x 9! 3!(9 -3)!

permutation A permutation is a selection of items where order is important The permutation of n objects taken r at a time is: P(n, r) = n! (n-r)!

example The permutation of 6 objects taken 3 at a time is: P(6, 3) = 6! (6 -3)! 3! = 6 x 5 x 4 x 3! = 6 x 5 x 4=120 3!

example The permutations of 3 letters (A, B, C) arranged into groups of 3 would be P(3, 3)= 3! = 3 x 2 x 1 =6 (3 -3)! 0! 1

Repeating letters When letters are repeated, permutations are not distinguishable from each other. Example: for the letters (B, O, B) There are only 3 distinguishable permutations- BOB, BBO, OBB Or the number of distinguishable permutations of n objects where one object is repeated q 1 times, another object is repeated q 2 times, and so on is n! q 1! q 2! …. . qk! So BOB = 3! = 3 2! 1!

Find permutations 1) find the permutation of 8 objects 2) find the permutation of 6 objects taken 2 at a time 3) find the permutation of 9 objects 4) find the permutation of 12 objects taken 3 at a time

Find the number of distinguishable permutations In the word CONNECTION It has 10 letters with C repeated 2 times, and N repeated 3 times and O repeated 2 times 10! = 151, 200 2! 3! 2!

combination A combination is a selection of items where order does not matter Combinations the combination of n objects taken r at a time is: C(n, r) = n! r!(n-r)!

example The combination of 5 objects taken 2 at a time is : 5! = 10 2!(5 -2)! 2!3!

Combination or permutation? Ask yourself if order is important. Example: the arrangement of digits for an alarm system requires a specific order. So order is important =PERMUTATION An arrangement of names in a class does not require a specific order, so order is not important = COMBINATION

Finding combinations Find the combination of 7 objects taken 4 at a time. Find the combination of 10 objects taken 3 at a time How many different pizza varieties can be created if there are 2 choices of crust, thick or thin, and 2 out of 3 possible toppings are selected. crust = C(2, 1) toppings = C(3, 2) C(2, 1) x C(3, 2) = ?

Independent or dependent events If event A and event B can both occur, then multiply their outcomes If event A or event B can occur, then add their events

hint When each combination represents a separate outcome, add the combinations

examples To travel from New York to Boston you could take one of three trains or one of four buses. How many different options for traveling between the two cities do you have?

Distinguishing between permutations and combinations Explain whether the following are permutations or combinations: 1) how many 2 letter arrangements can be formed from the letters CAT? order makes a difference because CA and AC are not the same arrangement= permutation

example How many 2 man crews can be selected from the set of 3 males: (Tom, Joe, Harry) Order does not matter- the crew of Tom and Joe is the same as Joe and Tom so = combination

careful When dealing with lettersnever assume that reusing a letter is allowed unless it specifically says it in the problem

Lab 7 Calculating permutations and combinations

Use graphing calculator To calculate a permutation 18 P 5 press 18 to enter n, the total number of objects to choose from. The subscript n is always located on the left hand side of the general form n. P r press MATH, go to PRB, then go down to 2: n. Press ENTER, then press 5, then ENTER

To calculate a combination Follow same procedure as permutation except select n. Cr instead of n. Pr

practice Calculate 12 P 7 P 22 3 12 C 7 C 22 3