Binomial theorem
In: Mathematics
That’s difficult to ELI5 without a context of use. The binomial theorem is just a mathematical formula, there is not a lot of thing to explain apart from why it is true, and for that you need to know how to multiply and add mathematical expressions. So, I will try to ELI**15** what the binomial theorem says and why it is true.
Maybe I should first explain the name? It is called the “binomial theorem” because a “binomial” is something like “a^(3) * b^(12) “, so two variables ‘a’ and ‘b’ each of them at a certain power, and multiplied together. And since the formula of this theorem use a lot of those, it gave the name to the theorem.
You probably already know that (30+12)^(2) = (30+12)*(30+12) = 30*30 + 30*12 + 12*30 + 12*12. That’s simple computation. Using the fact that 30*12 = 12*30, and generalizing, we obtain:
(a+b)^(2) = a^(2) + 2*a*b + b^(2)
Use a similar computation, you obtain:
(a+b)^(3) = (a+b)*(a^(2) + 2*a*b + b^(2) ) = a*a^(2) + a*2*a*b + a*b^(2) + b*a^(2) + b*2*a*b + b*b^(2)
Which we can simplify in:
(a+b)^(3) = a^(3) + 3*a^(2) *b + 3*a*b^(2) + b^(3)
In fact, you can easily generalize this and see that, up to some coefficients we will write “?” for now:
(a+b)^(n) = a^(n) + ?*a^(n-1) *b + ?*a^(n-2) *b^(2) + … + ?*a^(2) *b^(n-2) + ?*a*b^(n-1) + b^(n)
In other words, the binomial theorem simply says that: **The expression (a+b)^n is equal to a weighted sum of binomials.**
So, what are those coefficients “?”? Since each of those coefficients are the coefficient of a binomial, they are called **binomial coefficients.** More precisely, the binomial coefficient associated to a^(n-k) *b^(k) is said to be the **binomial coefficient for (n,k)** written usually with a “n” on top of a “k”, or C(n,k).
Those coefficients have a lot of good properties, and a good way to explain “What is C(n,k)?” is the following:
Imagine you have “n” player available for a sport game (football, …), and the rules of this game is that you need exactly “k” players. So you have to chose “k” players among the “n” available players. How many possible team exists?
C(n,k) is exactly how many “chose k among n possibilities” exists.
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