Editing 216: Romantic Drama Equation

{{comic
| number    = 216
| date      = January 29, 2007
| title     = Romantic Drama Equation
| image     = romantic drama equation.png
| titletext = Real-life prospective-pairing curves over things like age can get depressing.
}}

==Explanation==
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.

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.

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 alluding that a "gay cast" would consider only gay pairs, and a "straight cast" would consider only straight pairs.

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.

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!

===Formulas===
The formulas may be derived as follows:

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.

Wolfram|Alpha: [https://www.wolframalpha.com/input/?i=x*%28n-x%29 x*(n-x)]

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.

Wolfram|Alpha: [https://www.wolframalpha.com/input/?i=n*%28n-1%29%2F2%2Bx*%28x-n%29 n*(n-1)/2+x*(x-n)]

==Transcript==
:TV Romantic Drama Equation (Derived during a series of 'Queer as Folk' episodes)
:[A table shows equations for possible romantic pairings in a TV show. The equation under "gay" is n(n-1) 2+x(x-n); the equation under "straight" is x(n-x).]
:x: Number of male (or female) cast members.
:n: total number of cast members.

:[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}}
[[Category:Line graphs]]
[[Category:Homosexuality]]
[[Category:Math]]
[[Category:Romance]]