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I know that 35 unique quadrilaterals can be formed from a heptagon by joining the vertices. I can work that out in a diagram. What I want to know is the math behind figuring this out. An equation that will work with any shape (octagon for instance), and/or how this equation is derived. Detailed explanation please.

Mathematics
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http://www.wolframalpha.com/input/?i=7+choose+4 4 vertices make quadrilaterals. Heptagon has 7 vertices. So what you are doing is choosing 4 vertices out of 7 in unique ways.
^ Combinations formula. Seems legit.
I do understand that, but I want to know how this can be solve with out the use of a calculator.

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Other answers:

Factorials?
choosing 'r' out of 'n' can be written as \[ \binom{n}{r} = {n! \over r! (n-r)!} \]
Give me another example of what we are talking about that does not involve polygons.
most common example:- In a classroom you have 7 students. You have choose 5 students for basketball team. In how many different ways can you choose students.
Very good example: \[\left(\begin{matrix}7 \\5\end{matrix}\right)=\frac{ 7! }{ 5!(7-5)!}\]
yep!!
84
Now is there a quicker way to figure factorials without the use of a calculator? Obviously, 15! is going to take time and be cumbersome to figure out manually. Is there a short cut?
the topic is "Permutation and Combination" Permutation = choose something in order Combination = choose but order does not matter. ----------------------------------------- like you have two benches and 10 students. Each bench can hold 5 students. Q1. In how many ways can students sit in first bench? Q2. In how many ways can you ARRANGE student in first bench?
THANK YOU! That is exactly what I was looking for in the original question. Sorry if I wasn't clear.
http://www.wolframalpha.com/input/?i=7+choose+5
But seriously. If you have to do some extensive factorial work without a calculator, is there a short cut to going for example; 15*14*13*12* . . . *3*2?
lol .. .no without calculator this is horrible.
usually in combinations, the top and bottom cancel out so that it makes thing a bit simpler. that's the only easy portion when dealing with large numbers.
That's what I thought. See the thing is I'm studying for my CBEST and I was told that you cannot use a calculator, but I was just taking a sample test online and there are all these factorial questions that I'm like "look, I don't want to cheat, but how am I supposed to manually figure out all of these without a calculator in the allotted amount of time?"
What do you mean the top and bottom cancel out?
sorry .. numerator and denominator. when 'r' is close to 'n'. you can cancel out numerators and denominators and figure it out easily without calc.
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also when 'r' is pretty close to 'n', you can do the same.
Oh I see, yes. Still, I'm thinking somewhere I got erroneous information. I'm think that either the CBEST does allow a calculator, OR there are not nearly that many questions that involve factorials. If anyone reading this actually has firsthand experience in taking the CBEST direct message me.

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