Cereal The Post Corporation once introduced the cereal,...

Cereal The Post Corporation once introduced the cereal, Create a CrunchTM, in which the consumers could combine ingredients to create their own unique cereal. Each box contained 8 packets of food goods. There were four types of cereal: Frosted Alpha Bits®, Cocoa Pebbles®, Fruity Pebbles®, and Honey Comb®. Also included in the box were four “Add-Ins”: granola, blue rice cereal, marshmallows, and sprinkles. a. What is the total number of breakfasts that could be made if a breakfast is defined as any one or more cereals or add-ins? b. If Sara Ouellette chose to mix one type of cereal with one add-in, how many different breakfasts could she make? c. If Kristen Schmitt chose to mix two types of cereal with three add-ins, how many different breakfasts could she make? d. If Matthew Pastier chose to mix at least one type of cereal with at least one type of add-in, how many breakfasts could he make? e. If Inga Moffitt’s favorite cereal is Fruity Pebbles®, how many different cereals could she make if each of her mixtures must include this cereal?

Cereal The Post Corporation once introduced the cereal, Create a CrunchTM, in which the consumers could combine ingredients to create their own unique cereal. Each box contained 8 packets of food goods. There were four types of cereal: Frosted Alpha Bits®, Cocoa Pebbles®, Fruity Pebbles®, and
Honey Comb®. Also included in the box were four “Add-Ins”: granola, blue rice cereal, marshmallows, and sprinkles.
a. What is the total number of breakfasts that could be made if a breakfast is defined as any one or more cereals or add-ins?
b. If Sara Ouellette chose to mix one type of cereal with one add-in, how many different breakfasts could she make?
c. If Kristen Schmitt chose to mix two types of cereal with three add-ins, how many different breakfasts could she make?
d. If Matthew Pastier chose to mix at least one type of cereal with at least one type of add-in, how many breakfasts could he make?
e. If Inga Moffitt’s favorite cereal is Fruity Pebbles®, how many different cereals could she make if each of her mixtures must include this cereal?

Key Concepts

Fundamental Counting Principle

This principle states that if there are a ways to do one thing and b ways to do another, then there are a * b ways of performing both actions sequentially. It is used to calculate the total number of outcomes when independent events or choices are combined, as seen when choices from different groups (like cereals and add?ins) are multiplied together.

Subsets and Power Sets

A power set is the set of all subsets of a given set, including the empty subset and the full set itself. In combinatorial problems where you need to choose any one or more items from a group, the number of possible selections can be found by calculating 2^n and then removing the empty set if selections must not be empty.

Combinations (n choose k)

Combinations refer to selections of items where the order doesn't matter. The notation 'n choose k' is used to denote the number of ways to choose k items from a set of n. This concept is crucial when determining how many unique groups of items (such as cereals or add-ins) can be chosen from a larger set.

Counting with Restrictions

Counting with restrictions involves calculating the number of outcomes when certain conditions must be met—for example, choosing at least one item or including a specific item in every selection. This concept is useful when certain combinations are mandatory or when exclusions (like not choosing nothing) need to be applied.

Transcript

00:01 Okay, this question asks us about a situation where we have four cereals and four add -ins, and it asks us different questions about the number of breakfasts people can make.

00:11 So part a says we have to include at least one cereal or add -in.

00:16 So the number of subsets possible with this is equal to, as we said, 2 to the n, or 2 to the 8th in this case, which is 256.

00:31 But not all of those subsets are valid breakfasts because we need at least one cereal or add -in.

00:40 So we need to subtract away the one subset where there's nothing.

00:46 So we get 255 for the number of breakfasts.

00:52 Then for b, it says one cereal and one add -in.

00:58 So we have a cereal choice where we have four options, and then we have an add -in choice where we have four options.

01:07 So it's just 4 times 4, which is 16.

01:12 Then for part c, we have two serials and three add -ins.

01:21 So we have four cereals, and we're picking two of them, times we have four add -ins, and we're picking three of those.

01:30 So it's just four choose two times four choose three, which gives us 24 different options for this one.

01:44 Then for part d, it says we need at least one cereal and at least one add in.

01:53 So i'd start by having the 256 total potential breakfasts minus the ones we don't need...

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