P ERMUTATIONS AND C OMBINATIONS Homework: Permutation and Combinations WS

WARM UP There are 7 green marbles, 4 red marbles, and 2 blue marbles in the bag. Jenny picked a green marble from the bag, without replacement. What is the probability that the next marble picked is also green?

WARM UP- SOLUTION There are 7 green marbles, 4 red marbles, and 2 blue marbles in the bag. Jenny picked a green marble from the bag, without replacement. What is the probability that the next marble picked is also green? 6/12 or 1/2

COMBINATIONS Combination- order doesn’t matter. If you are dealt 5 cards from a deck it doesn’t matter what order you get them, when you pick up your hand you have 1 combination of cards. A combination is a grouping of the elements from a set in which the order doesn’t matter. In a combination, abc and acb would be considered the same: The elements are the same in both groups, and the order in which they appear does not matter.

EXAMPLE 1 How many combinations are there of the letters a, b, c and d using all letters? How many combinations are there using 3 of the letters?

EXAMPLE 1- SOLUTIONS How many combinations are there of the letters a, b, c and d using all letters? There is 1 combination. How many combinations are there using 3 of the letters? abc, abd, acd, bcd There are 4 combinations of 4 letters taken 3 at a time.

EXAMPLE 2 How many combinations are there of the 4 letters a, b, c and d using 2 letters at a time?

EXAMPLE 2- SOLUTION How many combinations are there of the 4 letters a, b, c and d using 2 letters at a time? ab ac ad bc bd cd There are 6 combinations.

EXAMPLE 3 How many combinations are there of the 4 letters a, b, c and d using 1 letter at a time?

EXAMPLE 3- SOLUTION How many combinations are there of the 4 letters a, b, c and d using 1 letter at a time? a b c d There are 4 combinations.

COMBINATIONS- FORMULA The combination of n things taken r at a time is 5! is read “five factorial”. It means (5)(4)(3)(2)(1) = 120

EXAMPLE 4 Find 10 C 6 There are lots of factors that you can cross out once you expand your factorials.

EXAMPLE 5 Find 6 C 2, 9 C 4 and 10 C 7. 6 C 2 = 9 C 4 = 10 C 7 =

EXAMPLE 5- SOLUTIONS Find 6 C 2, 9 C 4 and 10 C 7. 6 C 2 = 9 C 4 = 10 C 7 =

EXAMPLE 6 There are 6 questions on Elizabeth’s essay test. She only needs to answer 2 of them, she can choose any 2 that she wants. How many different combinations of 2 test questions can Elizabeth answer?

EXAMPLE 6-SOLUTION There are 6 questions on Elizabeth’s essay test. She only needs to answer 2 of them, she can choose any 2 that she wants. How many different combinations of 2 test questions can Elizabeth answer? 6 C 2 =

PERMUTATIONS A permutation is an arrangement of objects in an specific order. Order matters. $125 is very different that $512

EXAMPLE 7 How many permutations are there using the letters ABC?

EXAMPLE 7- SOLUTIONS How many permutations are there using the letters ABC? ABC, ACB, BCA, CBA, BCA, BAC = 6 These are dependent events, and using the fundamental counting principle we get 3 x 2 x 1 or 3!

PERMUTATIONS- FORMULA The permutations of n things taken r at a time is

EXAMPLE 8 Find 6 P 2, 9 P 4 and 8 P 5. 6 P 2 = 9 P 4 = 8 P 5 =

EXAMPLE 8- SOLUTIONS Find 6 P 2, 9 P 4 and 8 P 5. 6 P 2 = 9 P 4 = 8 P 5 =

EXAMPLE 9 Determine if each is a permutation or a combination. Assuming that any arrangement of letters forms a 'word', how many 'words' of any length can be formed from the letters of the word MATH? Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter.

EXAMPLES-SOLUTIONS Determine if each is a permutation or a combination. Assuming that any arrangement of letters forms a 'word', how many 'words' of any length can be formed from the letters of the word MATH? Permutation Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter. Combination How many ways are there to select a subcommittee of 7 members from among a committee of 17? Combination