I have a really simple question. I can't figure out the meaning of the binomial coefficient in the case of a binomial distribution formula. I know what the formula means, and how to use it for the combination but I can't link visually what it means in the case of the binomial distribution. I try to give an example. Imagine I have this 4 elements {a,b,c,d}, I want to take 2 element between 4, so $C_2^4$. I got this {ab,ac,ad,bc,bd,cd}, I have understood this formula, how it works and I understood his demonstration, here I'm 100% ok. But when I have to apply this to a case of TRUE and FALSE, like in the case of binomial distribution I can't see why it works. Exemple: I want 2 true and 2 false, I can take (v,v,f,f),(v,f,v,f),(f,v,v,f),(f,f,v,v),(f,v,f,v),(v,f,f,v). Here for the number of configuration I use the same formula. It's like to say:
- (a,b) -- (v,v,f,f)
- (a,c) -- (v,f,v,f)
- (a,d) -- (f,v,v,f)
- (b,c) -- (v,f,f,v)
- (b,d) -- (f,f,v,v)
- (c,d) -- (f,v,f,v)
The formula is the same, so there is something similar, but I can't find it, someone can explain me how te formula works in the case of true and false? Thx
Label the positions in your strings of $v$s and $f$s by the numbers $1$, $2$, $3$, $4$. So you get a mapping between strings and sets of $v$ positions, as shown in the table. $$ \begin{array}{c|c} 1234 & \text{$v$ positions}\\ \hline vvff & \{1,2\}\\ vfvf & \{1,3\}\\ vffv & \{1,4\}\\ fvvf & \{2,3\}\\ fvfv & \{2,4\}\\ ffvv & \{3,4\}\\ \end{array} $$