Lesson 7 So Many Possibilities! Develop Understanding
Jump Start
At Blaise’s Barbecue, they offer a meat plate with a choice of sides and bread. For the meats, customers can choose either beef brisket or pork ribs. Customers get to choose one of three sides: potato salad, coleslaw, or macaroni and cheese. The bread choices are either cornbread or a roll.
How many possible meat plates can be created with these choices? Show why your answer is correct.
Learning Focus
Represent the number of possible outcomes for a situation using diagrams, charts and formulas.
Compute probabilities using strategies.
How many ways are there to group a set of items? Are the number of possibilities different if the order matters?
Is there a way to figure out the possible groupings when there are too many items to write out all the possibilities?
How do the number of possible outcomes affect the probability?
Open Up the Math: Launch, Explore, Discuss
Blaise is working on setting up a new counter so that customers can see the food and select their choices. He knows he has 7 different items: brisket (B), pork ribs (P), coleslaw (CS), potato salad (PS), macaroni and cheese (MC), cornbread (CB), and rolls (R). Blaise wonders how many possible ways he could line up the seven items behind the counter.
1.
Help Blaise by finding the total number of ways that the seven items can be ordered along the counter. (Some of the possible arrangements may not make sense for Blaise’s restaurant, but we’ll let him figure that out. Our job is just to find the possibilities.)
2.
Blaise decides that he really has room for only 5 items along the counter. How many ways can 5 of the 7 items be placed in order?
3.
If Blaise decides to put the ribs and brisket in the first two spots, how many possible arrangements along the counter are there for the remaining positions?
4.
Besides the three side orders already mentioned, Blaise offers 2 others that can be chosen for an additional amount. Just to make things easier, we’ll call the 5 different sides, A, B, C, D, and E. (You get to imagine your favorites for the last two dishes.) For any combo meal, customers can choose 2 of the 5 different sides, with no repeats. How many different combinations of two sides are possible? Justify your answer.
5.
One of the regular customers loves all the sides at Blaise’s Barbecue. When she orders, she just chooses randomly (using a random number generator on her phone). If the side dishes are selected randomly, what is the probability that her combination will be
6.
What is the probability that a randomly selected combination includes either
7.
As he fills customer orders, Blaise wonders at the possibilities. He starts to think about the family meal deal. A customer can choose 2 from a selection of 6 meats, 3 side dishes from a selection of 5, 2 desserts from a selection of 7, and 1 bread from a selection of 2. How many different family meals are possible?
Ready for More?
At Quinn’s Super Q, they offer a mega-meal where customers can choose 3 sides from a menu of 7. We’ll call the sides A, B, C, D, E, F, G. Customers are not allowed to pick the same side twice. What’s the probability that a randomly selected combination will not include A?
Takeaways
The Fundamental Counting Principle:
Permutations | Combinations |
|---|---|
Formula for | Formula for |
Adding Notation, Vocabulary, and Conventions
Factorial:
Important Note:
Vocabulary
- combination
- factorial
- permutation
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we represented situations with tree diagrams, and other visual methods to count the possible outcomes for a situation. We applied the Fundamental Counting Principle and learned about two different ways to count possible outcomes, permutations and combinations. Permutations are used when the order of the items selected matters. Combinations are used when the order of the items in the group does not matter.
1.
Nyoka had half of a pecan pie remaining when two of her friends dropped by. She offered to divide the pie equally between the three of them.
a.
How much of the whole pie did each person get?
b.
What two fractions were multiplied to get the answer?
c.
When two fractions are multiplied together, is the product always smaller than either of the fractions being multiplied? Explain.
2.
Solve for