Combinatorics Chapter 3 Section 3.3 Permutations Finite Mathematics – Stall

3.2 IP Quiz

Review of Permutations Permutations – Ordered arrangements – the number of possible arrangements for an experiment in which the order of selections is important. “n objects taken n at a time” or “n objects taken k at a time”. No Replacement (distinguishable) P (n, n) = n! Example: _______________________________ P (n, k) = n!/(n-k)! Example:___________________________ With indistinguishable objects P(n,n) = n! /(n 1 ! +n 2 ! + …+ n k !) ______________________________ With Replacement P (n, k) = n k Example:__________________________________

Combinations Combinations – Groupings when order of selection does not matter – “A combination of n objects taken k at a time.” You select k objects from a set of n objects. The k objects can be selected in any order. Compare: Let us compare the number of permutations and the number of combinations that can be constructed of 4 objects taken 3 at a time. What is the Permutation formula? _____________________________

Combinations How do we calculate Combinations? We divide the permutation by k! It looks like this…

Combinations Example A: A group of ten people enter a contest with 3 color television sets as prizes. Each contestant’s name has been placed into a hat from which three names will be drawn as winners. How many different ways can the three winners be selected? How is this different than three people winning first prize, second prize and third? _________________________________________________ Let’s find the Combinations of 10 objects taken 3 at a time…

Combinations Work problems 1-4 on Page 82

Combinations Example B: A six member volleyball team is to be selected from a class of 6 boys and 8 girls. How many different ways can the team be selected if the number of boys must equal the number of girls? Solution: How many girls are there on the selected team? _______ How many boys are there on the selected team? _______ This is a two stage experiment and since the order of the selection is not important then we have a combination problem. (ways to select the boys) ∙ (ways to select the girls) C (6,3) ∙ C(8,3)=

BellWork 1. Evaluate P(10,3) 2. Evaluate 7! 3. Evaluate C(10,6) 4. How many seating arrangements exist for a round table with 6 seats? 5. How many different ways can a student answer 10 T/F questions?

Combinations What happens to our problems if the words “at least” or “at most” are contained in it? Example C: An urn contains 3 red (R), 1 white (W), and 2 blue (B) marbles. How many ways can 3 marbles be selected so that at least one red and at least one blue marble is selected? Think of the different scenarios possible with “at least one red and one blue” (1R, 1W, 1B) or _____________ or _______________ Then add the different results! The first combination is found like this: C(3,1) ∙ C(1,1) ∙ C(2,1) = 3 ∙ 2 ∙ 1 = 6 Find the other two ____________________________________________ ________________________________________________ Add 6 + _____ + ______ = ___________ This is how many ways that 3 marbles from the urn can be selected so that at least one is red and one is blue.

Combinations Work Problems 5-8

Pascal’s Triangle In Exercise 8 did you notice a pattern in the values of combinations before you added them? This is a well-known pattern that creates the triangle called Pascal ’ s Triangle. This can be a convenient way to obtain values of the C(n,r). What is C(7,3) = _________________

Work problems 9 and 10 Combinations

Example D: Five cards drawn form a deck of cards (52). How any different two pair hands can be selected? Solution: Identify the different stages that are in this experiment. Stage 1: Stage 2: Stage 3: Stage 4: Final equation will be C(13,3) ∙ C(3,2) ∙ C(4,2) ∙ C(4,2) ∙ C(4,1) = _____________________________________

Combinations One more Scenario… Let us consider the special case in which a set of objects are partitioned into subsets of the same size, but the order of the subsets is not important. Example: A class of 6 students is to be divided into two groups of three students for a field trip. Let S = {a,b,c,d,e,f}. Then the possible grouping of students would be For larger problems of this type we would use a formula…

Combinations Example E: How many ways can a teacher divide her class of twenty students into five study groups of four students each? Solution: n = _____ m = ____ k = _____ With the formula we get…

3.3 IP page 87-88: problems #15-29 odds. Combinations

Exit Quiz 1. Evaluate C(10,6) 2. Is the following statement T or F? C(5,3) = C(4,2) + C(4,3) 3. How many ways can 6 contracts be awarded to 10 different firms, if no two contracts are awarded to the same firm? 4. How many ways can the letters in HERRON HIGH SCHOOL be arranged?