As both theorems involve Pascal's Triangle, I first must display a theoretical output of the underlying premise:

As most know, the triangle consists of ways to determine (amongst other things) the binomial theorem, 2 to the nth power, and so on. However, what I discovered was merely a precursor to the majority of findings, such as why every combination has a pair, how many pairs per row, and, more specifically, the true root of the binomial theorem.

Again, what I found may seem so simplistic and rudimentary, however, as I stated, this had (as of a few months ago) not even been thought of/found. Here are the two theorems, in addition to my notes:

The math is rather simplistic on this, as we start with something as simple as knowing the row we are in. As the recursive definition states, A_{0} = .5 (signifying the zero row, or the top row of Pascal’s Triangle), we are left with a rather simple recursion in the note of A_{n} = A_{n} * 1/2, where n signifies the row, and for every integer, you have a logical number of pairs, and every fractional half, you have the whole number of pairs and the simple “pair” upon itself. However, there’s another feature to finding the amount of pairs within each row of Pascal’s Triangle.

Since Pascal’s Triangle uses combinations to find each value of the binomial theorem, we discover by adding the placement of the column, i.e. 0, 1, 2, 3, 4, 5 for the 6^{th} row, we discover by using the formula n(n-1), divided by the summation of the columns, we discover 2, as in the basis for the famous triangle and the theorems therein.

Using our simple example (the 6^{th} row), we discover n(n-1) = 6*5, or 30, and then divide it by the summation of the column numbers (0+1+2+3+4+5 = 15), we discover the number 2. The proof goes as follows:

N(N-1)/(N(N-1)/2) => 2(N(N-1))/(N(N-1)) => 2/1 => 2. First step, we multiply by the reciprocal of ½, so as to allow for a whole denominator. We are then left with 2 times the quantity of N times N minus 1, divided by the quantity of N times N minus 1. Since the quantity N times N minus 1 is divided by a quantity of the exact same measure, we are left with 2 divided by 1, or the number 2.

Or, rather, _{0}C_{5}, _{1}C_{5}, _{2}C_{5}, _{3}C_{5}, _{4}C_{5}, _{5}C_{5} translates to 5+5+5+5+5+5 over 0+1+2+3+4+5, for the value of 2.