Binomials theorem help~ (1 Viewer)

[Fiction]

I was wondering for the substitution one, cambridge mentions
(x+y ) ^n = n E k=0 nCk x^(n-k) y^k where E is sigma and the letter in front of it is the one on top, and the one just behind is the one drawn on the bottom

is that a general formula or only applicable to binomial theorems?
Also, again, cambridge mentions in the substitution theorem that substituting x =1, y=1 gives 2^n = n E k=0 nCk

It then says that, that simply means that the sum of every row is 2^n.

By the statement above, is it saying that every row of pascal's triangle is 2^n, therefore n has no set value? Like in row 3, it'll be 1 2 1 therefore n is 2? But in row 4, pascal's triangle is 1 3 3 1 so n is 3?

akfdadfs
Thanks in advance

 

[InteGrand]

I was wondering for the substitution one, cambridge mentions
(x+y ) ^n = n E k=0 nCk x^(n-k) y^k where E is sigma and the letter in front of it is the one on top, and the one just behind is the one drawn on the bottom

is that a general formula or only applicable to binomial theorems?
Also, again, cambridge mentions in the substitution theorem that substituting x =1, y=1 gives 2^n = n E k=0 nCk

It then says that, that simply means that the sum of every row is 2^n.

By the statement above, is it saying that every row of pascal's triangle is 2^n, therefore n has no set value? Like in row 3, it'll be 1 2 1 therefore n is 2? But in row 4, pascal's triangle is 1 3 3 1 so n is 3?

akfdadfs
Thanks in advance

The sum of entries in row of Pascal's Triangle is , where the topmost row corresponds to .

So for example, for row THREE (1 3 3 1, since row ZERO is the topmost row), the sum of entries is .

 

[InteGrand]

is that a general formula or only applicable to binomial theorems?

What do you mean by this? The formula IS the binomial theorem.