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==Explanation==
 
==Explanation==
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In a group of n people, such as the cast of a TV romantic drama like {{w|Queer as Folk (U.S. TV series)|Queer as Folk}}, the number of possible different pairs of people is n(n-1)/2. A romantic drama will often consider, over time, many possible romantic pairs of its cast members, even seeming to test the limit of how many pairs are possible. Through an austerely binaristic lens, this comic explores the implication of sex and sexual orientation in characterizing the possible pairs.
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The equations in the comic and the graph show how many different love pairs can be made if you know the number of females and males in a group. The text explains that it was inspired by TV Romantic Drama (in this case, the gay drama {{w|Queer as Folk (U.S. TV series)|Queer as Folk}}), but of course the formula is valid for any group of people. There are two graphs and equations - gay option is the case when we are looking for pairs with same gender, straight option is for heterosexual pairs. The interesting/funny part about the results is that in most cases there are more possibilities when we consider the homosexual option. Also it is interesting to observe what is kind of obvious - in the heterosexual case, the "best" case is if both genders are present equally, and the possibilities drop very fast if there is substantial difference between genders.
  
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If everyone in the group is male or female, then each pair could be classified as gay or straight. The formulas in the comic give how many of the possible pairs are gay and how many are straight, as a function of the total number of people and how many are male (or, equivalently, how many are female.) For example, of the 9 principal cast of Firefly, 5 are men and 4 are women. With n=9 and x=5, we have 16 possible gay pairs and 20 possible straight pairs.
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The graph makes a note that it only holds true for large casts. Case in point, with a cast of only four people: a two-to-two female-to-male ratio will have four straight pairings to two gay pairings, while a three-to-one female-to-male ratio will have three straight pairings and three gay pairings.
  
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A graph shows how the relative number of males and females affects the number of gay pairs and straight pairs. When the group is all male (or all female), all of the possible pairs are gay, but as the minority sex's number is increased, more of the pairs are straight. When the group is half male and half female, the number of straight pairs is maximized, and straight pairs slightly outnumber gay pairs. The curves are labeled "gay cast" and "straight cast", perhaps implying that a "gay cast" would consider only gay pairs, and a "straight cast" would consider only straight pairs.
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The title-text mentions that Randall made a chart of his own prospective dating pool as he gets older and was depressed by the results. A dating pool show how many possible partners you can choose from. As you get older, fewer of these will be single. But as he later shows in [[314: Dating Pools]], the age is not the problem... He is!
  
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There is a note that the graph describes large casts. Because all the quantities involved are discrete, for a small [n] there are only a few points to plot on the graph, and the smooth, continuous curves seen in the comic are less recognizable.
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===Formulas===
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The formulas may be derived as follows:
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Each straight couple needs to include one of the x males and one of the (n-x) females so there are x(n-x) possible ways of combining one of each.  E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and three possible pairings involving the second, yielding 2(5-2)=6 possible pairings.
  
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The title-text mentions that Randall made a graph of his prospective dating pool over time and was depressed by the results. As he gets older, his dating pool gets smaller: fewer people his age are single. But as Randall later shows in [[314: Dating Pools]], age is not the problem--he is!
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Each gay couple needs to include either two males or two females.  To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males.  However, that counts each possible pairing twice.  E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second.  To avoid double counting the possible couples, we therefore need to divide that total by 2.  So there are x(x-1)/2 possible male-male pairings.  Similar reasoning involving the (n-x) females tells us that there are (n-x)(n-x-1)/2 possible female-female pairings. Multiplying these out and combining the male and lesbian couples together, we get the total number of possible gay couples as [x^2 - x  +  n^2 - nx - n - xn + x^2 + x]/2.  That simplifies to [n^2 - n  +  2 x^2 - 2 xn]/2.  The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0].  Since the sum of these terms was divided by 2, we get that the total number of possible same-sex pairs is n(n-1)/2 - x(n-x), or n(n-1)/2 + x(x-n), which is what the cartoon says.
  
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===Formulas===
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===Equations and links to wolfram-alpha===
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The formulas may be derived as follows:
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* Formula for Gay pairing: [https://www.wolframalpha.com/input/?i=n*%28n-1%29%2F2%2Bx*%28x-n%29 n*(n-1)/2+x*(x-n)]
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* Formula for Straight pairing: [https://www.wolframalpha.com/input/?i=x*%28n-x%29 x*(n-x)]
  
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Each straight pair needs to include one of the x males and one of the (n-x) females, so there are x(n-x) possible ways of combining one of each.  E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and 3 possible pairings involving the second, yielding 2(5-2)=6 possible pairs.
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As a worked example, for Firefly, of the 9 principal cast, 5 are men and 4 are women. If all were gay, there would be 16 possible hook-ups; if all were straight, there would be 20.
  
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Wolfram|Alpha: [https://www.wolframalpha.com/input/?i=x*%28n-x%29 x*(n-x)]
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===Assumptions===
  
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Each gay pair needs to include either two males or two females.  To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males.  However, that counts each possible pair twice.  E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second.  To avoid double-counting the pairs, we therefore need to divide the product by 2.  So there are x(x-1)/2 possible pairs of two males.  Similarly, there are (n-x)(n-x-1)/2 possible pairings of two females. Summing these, we get the total number of possible gay pairs as [x^2 - x  +  n^2 - nx - n - xn + x^2 + x]/2.  That simplifies to [n^2 - n  +  2 x^2 - 2 xn]/2.  The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0].  Since the sum of these terms was divided by 2, we get that the total number of possible gay pairs is n(n-1)/2 - x(n-x), or n(n-1)/2 + x(x-n), which is what the cartoon says.
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The chart and the calculations assume that:
  
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Wolfram|Alpha: [https://www.wolframalpha.com/input/?i=n*%28n-1%29%2F2%2Bx*%28x-n%29 n*(n-1)/2+x*(x-n)]
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#No one is bisexual, asexual, polyamorous, celibate, or of some other sexuality.
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#The ENTIRE cast, male AND female, will ALL be of the same sexuality (all homosexual OR all heterosexual).
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#All genders are constant. (If sex-change operations are allowed, then the number of possible pairings is n(n-1)/2, because any two individuals can undergo surgery to become a possible pair.)
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#Gender is the only criterion for determining if a hook-up is possible.
  
 
==Transcript==
 
==Transcript==
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:n: total number of cast members.
 
:n: total number of cast members.
  
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:[A graph plots pairings (for large casts) against cast makeup. Each of the above equations forms a curve. "Gay cast" starts high for an all male cast, dips down at 50/50 cast makeup, and then rises again for all female. "Straight cast" starts at zero for an all male cast, peaks at 50/50 cast makeup, and then drops to zero again for an all female cast. The two curves intersect at two points close to the middle.]
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:[A graph plots pairings (for large casts) against cast makeup. Each of the above equations forms a curve. "Gay cast" starts high for an all male cast, dips down at 50 50 cast makeup, and then rises again for all female. "Straight cast" starts at zero for an all male cast, peaks at 50/50 cast makeup, and then drops to zero again for an all female cast. The two curves intersect at two points close to the middle.]
  
 
{{comic discussion}}
 
{{comic discussion}}

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