MDM4UI UNIT 2 Day 2 Combinations
In the last unit, we studied permutations. In a permutation, there is a difference between selecting, say, Bob as president and Margot as vice-president as opposed to Margot as president and Bob as vice-president. If you select Bob and Margot as co-chairs, the order in which you select them does not matter since they will both have the same job.
COMBINATION: A selection from a group of objects without regard to the order. The number of combinations of r objects chosen from a set of n items is
nCr=C(n,r)=nr=n!(n-r)!r!
This is read as “n choose r”
PERMUTATION: A selection from a group of objects where order matters.
Comparing Permutations and Combinations
Example 1:
(a) In how many ways could Alana, Barbara, Carl, Domenic, and Edward fill the positions of president, vice-president, and secretary?
(b) In how many ways could these same five people form a committee with three members? List the ways.
(c) How are the numbers of ways in parts (a) and (b) related?
Applying the Combinations Formula
Example 2: How many different sampler dishes with 3 different flavours could you get at an ice-cream shop with 31 different flavours?
Calculating Numbers of Combinations Manually
Example 3: A ballet choreographer wants 18 dancers for a scene. In how many ways can the choreographer choose the dancers if the company has 20 dancers? 24 dancers?
Using the Counting Principles with Combinations
Example 4: Ursula runs a small landscaping business. She has on hand 12 kinds of rose bushes, 16 kinds of small shrubs, 11 kinds of evergreen seedlings, and 18 kinds of flowering lilies. In how many ways can Ursula fill an order if a customer wants
(a) 15 different varieties consisting of 4 roses, 3 shrubs, 2 evergreens, and 6 lilies?
(b) either 4 different roses or 6 different lilies?
Practice
1. Explain why n objects have more possible permutations than combinations. Use a simple example to illustrate your explanation.
2. Explain whether you would use combinations or permutations to calculate the number of ways of choosing:
a. Three items from a menu of ten items.
b. An appetizer, an entrée, and a dessert from a menu with three appetizers, four entrées, and five desserts.
c. You have 100 songs on your iPod. In how many ways can you listen to 15 different songs?
3. Convert to factorial form, then evaluate.
a. 9C5 b. 8C4 c. C(12,3) d. 115
e. 7C2 x 6C3
4. In how many ways could you choose 4 packages of pasta from a bin containing 11 different packages of pasta?
5. On an English exam, students need to answer six out of eight questions in Part A and two out of four questions in Part B. The order in which they answer the questions does not matter. In how many ways could a student answer the questions on this exam?
6. Juries are chosen from large pools of people selected at random from the local population. A jury pool has 40 people.
a. How many ways are there to form a 12-person jury in a criminal case?
b. How many ways are there to form a 6-person jury in a civil case?
c. Which situation gives a larger number of ways? Explain why this is to be expected?
7. A dealership has six models of trucks and five models of cars for sale. Wayne sells four vehicles this week. How many of the following combinations of four can be formed?
a. No restrictions
b. Two trucks and two cars
c. Three cars and one truck
d. Only cars
e. Only trucks
f. How are the answers to parts b. to e. related to part a.? Explain why this is true?
8. Ten identical playing pieces are placed on a 5x5 game board.
a. In how many ways could 10 playing pieces be placed
on the board if there are no restrictions?
b. In how many ways could 10 playing pieces be placed on
the board if there must be two pieces in each row?
c. Describe how the results would change if the playing pieces
were all different.
Textbook: Pg. 262 #1, 2, 3, 5, 6