---
tags: 225-spr22, mth225, lt-quiz
---

# Mini-Quiz June 14

:::info
This is a **mini-quiz**, designed to give you an attempt at **a small number of Learning Targets** of your choice from among the more recent ones we've covered, especially CORE targets, done **quickly** and **in class**.

This mini-quiz contains problems for **Learning Targets 13, 14, and 21**. You may do any combination of these you want. Or if you prefer to practice some more and try again this week, do nothing and just take the main Quiz on Thursday.

Work the problems you select out on paper or in a notes app, export the work to PDF (one PDF per Learning Target) and upload it to the appropriate assignment folder on Blackboard just as in the case of a full Quiz.

**Your work is due by 11:59pm. Note the later due date this time!** No submissions will be accepted after that time.
:::

## Learning Target 13

:::warning
(**CORE**) I can apply the Additive and Multiplicative Principles and the Principle of Inclusion/Exclusion to formulate and solve basic combinatorics problems.
:::

*Work out the answer to each of the counting questions below. State the answer clearly, and show all work and explain your reasoning in words on each. **Note:** Answers with insufficient explanations or work are not accepted.*

1. It's the first day of summer camp, and the counselors are trying to figure out where one of the campers will sleep. There are four beds available in Cabin #1, five in Cabin #2, and just one in Cabin #3. How many ways in all are there to assign the camper to a bed?
2. Campers at the summer camp are allowed to go out on the lake in a boat as long as they are either 13 years old or older, or their parents signed an insurance waiver. The table below has some relevant information about the campers (50 of them in all):

| | Signed waiver | Did not sign waiver | *Total*
| --- | :---: | :----: | :-----: |
| **12 or younger** | 10 | 20 | 30 |
| **13 or older** | 8 | 12 | 20 |
| *Total* | 18 | 32 | 50 |

How many campers are allowed to go out on the lake in a boat?

**Success criteria:** All three parts have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least two of the three answers must be correct and sufficiently justified; the third may be incorrect as long as it results from a "simple" error.

## Learning Target 14

:::warning
(**CORE**) I can compute a binomial coefficient and apply the binomial coefficient to solve basic combinatorics problems.
:::

1. Compute the exact numerical value of the following. Either show work or explain in one sentence how you got your answer. Leave no fractions in your answers.
(a) $\binom{12}{5}$
(b) $\binom{33}{12}$

2. Solve the following counting problems. State your answer clearly and justify each answer with math work or a brief verbal explanation. The justification for each answer must involve the binomial coefficient.
(a) A psychology study requires a random sample of 12 people from a population of 24. How many possible selections of this size are there?
(b) How many 8-bit strings have either 5 `1` bits or 6 `1` bits in them?

**Success criteria:** All of the answers on the first item are correct and have sufficient justifications. Both parts of the second item have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least one of the two answers must be correct and sufficiently justified; the other may be incorrect as long as it results from a "simple" error.

## :new: Learning Target 21

:::warning
(**CORE**) Given a statement to be proven by mathematical induction, I can identify the predicate, prove the base case, state the inductive hypothesis, and sketch a proof.
:::

Consider the statement: **For any positive integer $n$, $n^3 + 2n$ is a multiple of $3$.** Set up the framework for a proof by mathematical induction by doing the following steps:

1. State the base case, and prove that the base case is true.
2. Clearly state the inductive hypothesis.
3. Clearly state the inductive step (that is, clearly state what needs to be proven following the inductive hypothesis).
4. Give at least one suggestion for how a full proof of the inductive step might go, that is reasonable and likely to be useful.

**Success criteria:** The base case is clearly stated and correctly proven. The inductive hypothesis is clearly and correctly stated. The inductive step is clearly and correctly stated.