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**COMBINATOIRCS**

**C1**
<br/>A group of students is surveyed about race and gender identity for research purposes. The survey lists 6 categories for race and 4 categories for gender. Determine how many different survey results are possible, given each of the following constraints:
<br/> A group of students is surveyed about race and gender identity for research purposes. The survey lists 6 categories for race and 4 categories for gender. Determine how many different survey results are possible, given each of the following constraints:
* Students are allowed to check exactly 1 box under race and 1 box under gender.
* Students are allowed to check 1 or 2 boxes under race and 1 or 2 under gender.
* Students are allowed to check as many boxes as they wish (including 0).

**C2**
<br/>A student club focused on diversity is forming a recruitment committee consisting of 7 of its 30 members. 10 of the club’s members identify as gay and 5 identify as asexual. How many ways are there to form a recruitment committee which includes at least three members from each of these groups?
<br/> A student club focused on diversity is forming a recruitment committee consisting of 7 of its 30 members. 10 of the club’s members identify as gay and 5 identify as asexual. How many ways are there to form a recruitment committee which includes at least three members from each of these groups?

**C3**
<br/>In how many ways can three men, three women, and three non-binary students stand in a circle if no two people of the same gender identity are allowed to stand next to each other?
<br/> In how many ways can three men, three women, and three non-binary students stand in a circle if no two people of the same gender identity are allowed to stand next to each other?

**C4**
<br/>You conduct an experiment in which you interview a large number of families, each of which has 5 children. For each family, you write down the biological sex (M for male, F for female, I for intersex) of the children, in order from oldest to youngest.
<br/> You conduct an experiment in which you interview a large number of families, each of which has 5 children. For each family, you write down the biological sex (M for male, F for female, I for intersex) of the children, in order from oldest to youngest.
* How many possible results are there?
* Out of all the possible results, how many have exactly two I’s?
* Out of all the possible results, how many have at least one I, one F, and two M’s?

**C5**
<br/>Jon wants to choose a different three-letter name for herself to better match her pronouns. How many consonant-vowel-consonant options does she have to choose from?
<br/> Jon wants to choose a different three-letter name for herself to better match her pronouns. How many consonant-vowel-consonant options does she have to choose from?

**C6**
<br/>Consider the set of letters, *{L, G, B, T, Q, I, A}*:
<br/> Consider the set of letters, *{L, G, B, T, Q, I, A}*:
* How many ways are there to form a string containing exactly one copy of each letter?
* Suppose all license plates in a certain state consist of three distinct consonants followed by two distinct vowels. How many such license plates can be formed from the given set?
* Suppose three letters are chosen from this set at random (repeats allowed); what are the odds that at least one of them is a vowel (rounded to four digits)?

**C7**
<br/>Charlie and her girlfriend, Amber, go to lunch at a restaurant with 7 sandwiches on the menu. There are also 5 beverages on the menu. How many possible two-sandwich, two-drink meals can they order (regardless of which of them orders first)...
<br/> Charlie and her girlfriend, Amber, go to lunch at a restaurant with 7 sandwiches on the menu. There are also 5 beverages on the menu. How many possible two-sandwich, two-drink meals can they order (regardless of which of them orders first)...
* if they order different sandwiches and drinks from each other?
* if they each order without concern for what the other ordered?

**C8**
<br/>A small civil rights non-profit is selecting 3 of its 15 employees to form a task force focused on voting rights.
<br/> A small civil rights non-profit is selecting 3 of its 15 employees to form a task force focused on voting rights.
* How many such task forces could they possibly select?
* How many possibilities are there if they assign each member of the task force a unique role?

**C9**
<br/>100 people are marching on the capitol to protest racial injustice. They intend to organize themselves into a pentagonal marching pattern, each point of the pentagon consisting of a four-person by five-person rectangular formation.
<br/> 100 people are marching on the capitol to protest racial injustice. They intend to organize themselves into a pentagonal marching pattern, each point of the pentagon consisting of a four-person by five-person rectangular formation.
* How many ways are there to separate the marchers into five groups?
* Within each group, how many possible four-by-five formations are there?
* How many ways are there to arrange five different objects to form a pentagon?
* Overall, how many possible organizations do the marchers have to choose from?

**C10**
<br/>A group of 12 highschoolers are attending queer prom together. If they want to arrange themselves into 6 couples, how many possible couple arrangements can they choose from?
<br/> A group of 12 highschoolers are attending queer prom together. If they want to arrange themselves into 6 couples, how many possible couple arrangements can they choose from?

**C11**
<br/>The organizers of a LGBTQ rally are making pride flags. Five people all choose different flag designs from among ten; how many possible combinations of flags might they end up with?
<br/> The organizers of a LGBTQ rally are making pride flags. Five people all choose different flag designs from among ten; how many possible combinations of flags might they end up with?

---------------------
**FORMAL PROOF**

**F1**
<br/>Suppose 100 highschoolers attend a queer prom. Use the Pigeonhole Principle to prove that at least 9 of them share the same birth month.
<br/> Suppose 100 highschoolers attend a queer prom. Use the Pigeonhole Principle to prove that at least 9 of them share the same birth month.

---------------------
**GRAPH THEORY**

**G1**
<br/>The graph below depicts connections in a blockchain network used to trade NFTs. Nodes represent individual users; an edge between two users means that they are contacts on the network. Thus, any walk along this graph represents a possible sequence of transactions for a hypothetical NFT.
<br/> The graph below depicts connections in a blockchain network used to trade NFTs. Nodes represent individual users; an edge between two users means that they are contacts on the network. Thus, any walk along this graph represents a possible sequence of transactions for a hypothetical NFT.
* Find a sequence of 5 transactions which involves someone selling the NFT before buying it back from the same person they sold it to.
* Is the walk you found above a trail? Why or why not?
* Is the walk you found above a cycle? Why or why not?

<img src="https://user-images.githubusercontent.com/59627709/120771925-fe1b8e00-c4dc-11eb-89c3-9578ebbd7c07.jpg" width="730" height="370">

**G2**
<br/> Robin has drawn two different family trees (labeled A and B below), both of which represent familial relationships between all seven members of their household. They live with their adoptive parents, Maya and Imani; their auncle, Kai; their grandparents, Carmen and Bo; and their great-grandmother, Ana.
* In tree A, what does a horizontal dashed line represent?
* In tree B, what does a horizontal solid line represent?
* In general, what's the difference between solid and dashed lines in these two trees?
* Why does Maya's node have three solid edges in tree B, but only one in tree A?
* Which representation of Robin's household do you prefer? Why?

<img src="https://user-images.githubusercontent.com/59627709/122342618-168aa000-cf02-11eb-8a7e-ec0a597848e3.jpg" width="730" height="370">
<img src="https://user-images.githubusercontent.com/59627709/122342712-299d7000-cf02-11eb-9d62-cf2fd8f650dc.jpg" width="730" height="370">

---------------------
**PROBABILITY THEORY**

**P1**
<br/>Three people are on a date together and decide to play an ice-breaking game in which they try to guess each other’s favorite seasons. (They write their guesses down all at once so the game isn’t spoiled when the answers are revealed.) What are the odds that:
<br/> Three people are on a date together and decide to play an ice-breaking game in which they try to guess each other’s favorite seasons. (They write their guesses down all at once so the game isn’t spoiled when the answers are revealed.) What are the odds that:
* At least one of their six guesses is correct?
* Exactly two of their guesses are correct?

**P2**
<br/>Imagine you attend a civil rights rally, looking for a friend who has volunteered their time to the event. You know that they will be at one of the 12 informational booths for half of the event, and another for the rest. If you only manage to visit 6 of the booths (dividing your time at each equally), what are your chances of finding them?
<br/> Imagine you attend a civil rights rally, looking for a friend who has volunteered their time to the event. You know that they will be at one of the 12 informational booths for half of the event, and another for the rest. If you only manage to visit 6 of the booths (dividing your time at each equally), what are your chances of finding them?

**P3**
<br/>A club of 200 students is forming a committee with 10 members. Given that 15% of club members identify as queer, what are the odds that a randomly selected committee reflects this ratio within a 10% margin?
<br/> A club of 200 students is forming a committee with 10 members. Given that 15% of club members identify as queer, what are the odds that a randomly selected committee reflects this ratio within a 10% margin?

**P4**
<br/>Suppose you attend a drag show in which two of your friends are performing. There are also two other performers scheduled that night, but you don’t know what order they’ll be in and you only have time to stay for half the show. What are the odds that:
<br/> Suppose you attend a drag show in which two of your friends are performing. There are also two other performers scheduled that night, but you don’t know what order they’ll be in and you only have time to stay for half the show. What are the odds that:
* You’ll see both of your friends perform?
* You’ll see just one of your friends perform?

**P5**
<br/>Imagine you are conducting an on-campus survey to measure student awareness of the legacy of colonialism and slavery. The survey consists of five true/false historical questions, and you survey six students. Suppose, for simplicity, that every student has a 50/50 chance of answering each question correctly. What are the odds that:
<br/> Imagine you are conducting an on-campus survey to measure student awareness of the legacy of colonialism and slavery. The survey consists of five true/false historical questions, and you survey six students. Suppose, for simplicity, that every student has a 50/50 chance of answering each question correctly. What are the odds that:
* A given student scores 100%?
* A given student scores at least 80%?
* Two or more students score at least 60%?
* Also, is it possible for all students to get different scores? Why or why not?

**P6**
<br/>Your local gay bar is hosting a trivia night, and you decide to test your psychic abilities by participating and answering each question blindly. Suppose you answer 5 multiple-choice questions - each with four possible answers. What are your odds of:<br/>
<br/> Your local gay bar is hosting a trivia night, and you decide to test your psychic abilities by participating and answering each question blindly. Suppose you answer 5 multiple-choice questions - each with four possible answers. What are your odds of:<br/>
* Answering none of the questions correctly?
* Answering three of the questions correctly?
* Answering three questions correctly in a row?

**P7**
<br/>A recent Stanford University study on racial bias in police traffic stops found that, among the US municipalities it analyzed, “the annual per-capita stop rate for black drivers was 0.20 compared to 0.14 for white drivers. For Hispanic drivers … 0.09”. Now, imagine a US city in which 40% of the population is black, 35% is white, and 25% is Hispanic. Working from the data provided by the aforementioned study, what is the approximate likelihood that a given driver stopped by the police in this city is black?<br/>
<br/> A recent Stanford University study on racial bias in police traffic stops found that, among the US municipalities it analyzed, “the annual per-capita stop rate for black drivers was 0.20 compared to 0.14 for white drivers. For Hispanic drivers … 0.09”. Now, imagine a US city in which 40% of the population is black, 35% is white, and 25% is Hispanic. Working from the data provided by the aforementioned study, what is the approximate likelihood that a given driver stopped by the police in this city is black?<br/>
[SOURCE](https://5harad.com/papers/100M-stops.pdf)

**P8**
<br/>Recent census data has concluded that approximately 1.44% of US couples living together are “same-sex”. Of these, about 53.66% are legally married, which accounts for roughly 0.88% of married households in the country. Given a randomly selected US couple living in the same household (of any sexual orientation), what is the approximate likelihood that they are married?<br/>
<br/> Recent census data has concluded that approximately 1.44% of US couples living together are “same-sex”. Of these, about 53.66% are legally married, which accounts for roughly 0.88% of married households in the country. Given a randomly selected US couple living in the same household (of any sexual orientation), what is the approximate likelihood that they are married?<br/>
[SOURCE](https://www.census.gov/newsroom/press-releases/2019/same-sex-households.html)

---------------------
**SET LOGIC**

**S1** *-pairs with Factoid #6-*
<br/>Imagine you've created an algorithm for automatically writting sappy love letters. Such an algorithm might work, sort of like the game MadLibs, by generating letter templates and replacing blank spaces with words drawn from collections of amorous terminology. For example, consider the template sentence, *S = "My LABEL, you have made me the most ADJECTIVE LABEL in the PLACE."* where *LABEL* is any member of the set *L = {dear, wife, husband, partner}*, *ADJECTIVE* is any member of the set *A = {enamored, overjoyed, fortunate}*, and *PLACE* is any member of the set *P = {world, galaxy, history of spacetime}*.
* *(a)* How many different possible variations of the given example sentence, *S*, could your algorithm produce?
* *(b)* How does your answer to part *(a)* change depending on whether we allow repeats?
<br/> Imagine you've created an algorithm for automatically writting sappy love letters. Such an algorithm might work, sort of like the game MadLibs, by generating letter templates and replacing blank spaces with words drawn from collections of amorous terminology. For example, consider the template sentence, *S = "My LABEL, you have made me the most ADJECTIVE LABEL in the PLACE."* where *LABEL* is any member of the set *L = {dear, wife, husband, partner}*, *ADJECTIVE* is any member of the set *A = {enamored, overjoyed, fortunate}*, and *PLACE* is any member of the set *P = {world, galaxy, history of spacetime}*.
* How many different possible variations of the given example sentence, *S*, could your algorithm produce?
* How does your above answer change depending on whether we allow repeats?

---------------------
---------------------
# Affirming CS/Math History Factoids
*For potential use in-place-of or alongside Affirming Problems*

**#1**
<br/>George Boole was husband to Mary Everest Boole, a self-taught English mathematician and teacher who lived during the 19th and 20th Centuries. As an advocate for educational methods which were progressive at the time, she used fables and stories to teach algebra. She also invented curve-stitching for similar purposes, which has since evolved into string art.
<br/> George Boole was husband to Mary Everest Boole, a self-taught English mathematician and teacher who lived during the 19th and 20th Centuries. As an advocate for educational methods which were progressive at the time, she used fables and stories to teach algebra. She also invented curve-stitching for similar purposes, which has since evolved into string art.

**#2**
<br/>The function f(x)=3(x^2)+5 describes a parabola. If you lived in Alexandria during the age of Rome, you might have learned about parabolas and other conic sections from Hypatia's exegesis on Apollonius' *Conics*. Hypatia was a Greek scholar and teacher, as well as the first female mathematician whose life is well-documented.
<br/> The function f(x)=3(x^2)+5 describes a parabola. If you lived in Alexandria during the age of Rome, you might have learned about parabolas and other conic sections from Hypatia's exegesis on Apollonius' *Conics*. Hypatia was a Greek scholar and teacher, as well as the first female mathematician whose life is well-documented.

**#3**
<br/>The Bernoulli number sequence was the subject of the first published algorithm, written by Ada Lovelace in 1840 to be run on Charles Babbage's Analytical Engine, an early version of the modern computer.
<br/> The Bernoulli number sequence was the subject of the first published algorithm, written by Ada Lovelace in 1840 to be run on Charles Babbage's Analytical Engine, an early version of the modern computer.

**#4**
<br/>Turing machines share their namesake with the Turing test, which has inspired science fiction stories from *Blade Runner* to *I, Robot*. Both are named after the 'father of artificial intelligence', Alan Turing, whom Queen Elizabeth II posthumously pardoned in 2013 - more than half a century after he was prosecuted for homosexual acts and sentenced to chemical castration by the UK government.
<br/> Turing machines share their namesake with the Turing test, which has inspired science fiction stories from *Blade Runner* to *I, Robot*. Both are named after the 'father of artificial intelligence', Alan Turing, whom Queen Elizabeth II posthumously pardoned in 2013 - more than half a century after he was prosecuted for homosexual acts and sentenced to chemical castration by the UK government.

**#5**
<br/>Alan Turing's work on the Halting Problem played a pivotal role in demonstrating the undecidability of mathematics. [This](https://www.youtube.com/watch?v=HeQX2HjkcNo) video discusses Turing's work on undecidability in the context of Kurt Godel's famous incompleteness proof; taken together, they raise profound questions about the nature of logic and reason.
<br/> Alan Turing's work on the Halting Problem played a pivotal role in demonstrating the undecidability of mathematics. [This](https://www.youtube.com/watch?v=HeQX2HjkcNo) video discusses Turing's work on undecidability in the context of Kurt Godel's famous incompleteness proof; taken together, they raise profound questions about the nature of logic and reason.

**#6** *-pairs with Problem S1-*
<br/> This problem was inspired by the [Strachey love letter algorithm](https://en.wikipedia.org/wiki/Strachey_love_letter_algorithm), written in 1952 for the Manchester Mark 1. Its author, Christopher Strachey was a british computer scientist during the mid-20th Century who pioneered the idea of time-sharing. He also developed an early version of checkers for the Manchester Ferranti.
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