Exercises

Exercises 7.11 Exercises

1.

Calculate the number of ways to choose (using combinations):

$2$ municipalities from a county with $10$ total.
$3$ municipalities from a county with $10$ total.
$3$ municipalities from a county with $11$ total.
$3$ municipalities from a county with $12$ total.
$3$ municipalities from a county with $n$ total (simplify the expression so that it contains no factorials).
$n-3$ municipalities from a county with a total of $n\text{.}$ Explain why it agrees with your answer in e.
$4$ municipalities from a county with $n$ total.
Explain why the formula for choosing $k$ municipalities from a county with $n$ total is a polynomial of degree $k\text{.}$

2.

Suppose OAFC wishes to place a vaccine clinic and a dental clinic in two separate municipalities of the $30$ in Walworth county. In how many ways can they do this?

3.

Suppose they wish to place a vaccine clinic, a dental clinic, and a check-up clinic in three separate municipalities of the $30\text{.}$ In how many ways can they do this?

4.

Suppose they wish to choose two clinic types out of the three: vaccine clinic, dental clinic, and check-up clinic, and then to place two clinics of the chosen types in two separate municipalities of the $30\text{.}$ In how many ways can they do this?

5.

(harder) Suppose they wish to place two vaccine clinics and one dental clinic in three separate municipalities out of $30\text{.}$ In how many ways can they do this? What about two vaccine clinics, one dental clinic, and three check-up clinics?

6.

Suppose OAFC wishes to open at least $1$ and at most $5$ clinics in separate municipalities out of the $30$ in Walworth county. In how many ways can they do this?

7.

Suppose OAFC wishes to open at least $2$ and at most $5$ clinics in separate municipalities out of the $30$ in Walworth county. In how many ways can they do this? Use the previous exercise and the subtraction rule.

8.

In section Section 7.9, the relationship between $y\text{,}$ 200% of the FPL in a fixed municipality, and $r\text{,}$ the fraction of how "far" $y$ is along the way from to $$25000$ to $$50000\text{,}$ was said to be given by

\begin{equation*} r=\frac{y-25000}{25000}. \end{equation*}

Explain this formula in your own words.

9.

Suppose a state-funded aid organization has modeled medical need in a state with $100$ counties. They have the resources to open as many as $20$ clinics in diffent counties. If they have access to a computer which can check the need covered for one billion different possible clinic placements, what is the maximum number of clinics for which they can compute all placement possibilities (for that number of clinics) exhaustively?

10.

Recall the formulas for $nCm$ and $nPm\text{:}$

\begin{equation*} nPm=\frac{n!}{(n-m)!} \end{equation*}

\begin{equation*} nCm=\frac{n!}{(n-m)!m!}\text{.} \end{equation*}

Explain how the formula for $nCm$ comes from the formula for $nPm$ and the division rule.

11.

Pascal’s triangle (perhaps more appropriately known as Khayyam’s triangle or Yang Hui’s triangle after those who studied it earlier) provides a method of computing the binomial coefficients (numbers of the form $nCk=\binom{n}{k}$) in an iterative way [7.12.1.128]. One starts with an infinite row of $0$s with a single $1$ in the middle:

\begin{equation*} \ldots 0 \quad 0 \quad 0 \quad 0 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \ldots \end{equation*}

Subsequent rows are computed by putting the sum of the two numbers above in the gap below them:

\begin{equation*} \ldots 0 \quad 0 \quad 0 \quad 0 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \ldots \end{equation*}

\begin{equation*} \ldots 0 \quad 0 \quad 0 \quad 0 \quad 1 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \ldots \end{equation*}

\begin{equation*} \ldots 0 \quad 0 \quad 0 \quad 0 \quad 1 \quad 2 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \ldots \end{equation*}

\begin{equation*} \ldots 0 \quad 0 \quad 0 \quad 0 \quad 1 \quad 3 \quad 3 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \ldots \end{equation*}

We can ignore the $0$s in order to draw it more compactly:

\begin{equation*} \begin{array}{c} 1 \\ 1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad 1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \\ 1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1 \\ 1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1 \\ 1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1 \\ \end{array} \end{equation*}

Now compute $\binom{4}{0}\text{,}$ $\binom{4}{1}\text{,}$ $\binom{4}{2}\text{,}$ $\binom{4}{3}\text{,}$ and $\binom{4}{4}$ using the formula from section Section 7.5 and compare the numbers you get to the triangle above. What do you notice? Make a conjecture (guess) about the binomial coefficients and the $5^\text{th}\text{,}$ $6^\text{th}\text{,}$ and $n^\text{th}$ row of Pascal’s triangle.
(requires algebra) Consider the expression $(x+1)^n\text{.}$
1. Expand $(x+1)^2$ by distributing and collecting like terms.
2. Expand $(x+1)^3$ by distributing and collecting like terms.
3. Make a guess which connects the coefficients (numbers appearing before the variables in the expanded expression) and Pascal’s triangle.
4. The sum of the entries in the $3^\text{rd}$ row of Pascal’s triangle is
  \begin{equation*} 1+3+3+1=8. \end{equation*}
  Compute the sum of entries in all eight of the rows shown. What patterns do you see?
5. In fact, if we call the top row the $0^\text{th}$ row, the sum of entries in the $n^\text{th}$ row is $2^n\text{.}$ Can you explain why? Hint: Set $x=1\text{.}$
Suppose that OAFC wants to consider the possibility of placing any number of mobile clinics (as few as $0$ and as many as $30$) in separate municipalities in Walworth county. How many possible ways can they do this?

12.

Arguably, a smarter model than the one we settled on would take into account the distance from each centroid to a clinic location and weight the need satisfied at this municipality according to this distance. Expand upon how this might be done to improve our solution.

13.

The map visualizations throughout this chapter were created by importing and analyzing publically available data from the American Community Survey using a python library called GeoPandas [7.12.1.143]. The author's code is also available [7.12.1.136]. The data was imported from GeoJSON files which can be obtained at the censusreporter.org links in the references, e.g. following the link at [7.12.1.140], clicking "Download data" at the top right and then GeoJSON in the dropdown menu. Follow the online YouTube tutorial at [7.12.1.119] to get this software up and running on your own machine, and follow the one at [7.12.1.126] to learn how to import and plot data from GeoJSON files. The more at advanced tutorial [7.12.1.122] may be helpful as well.