How the pattern emerging in the exponents of the terms of a binomial expansion is proven?

Question

Every article I've read just shows that the exponents of a term in the expansion are of the form $$a^{n-k}b^{k}$$ but how to prove that this is really true?

Answer 1

Using the principle of mathematical Induction we can sketch a proof like this:

1. Base case: n=1, $$(a+b)^1 = a^1+b^1 = a^1b^0 + a^0b^1$$
2. Assume true for case $n$: $$(a+b)^n = a^n + c_1 a^{n-1}b^1 + \cdots+ b^n$$
3. Induction step : investigate case $n+1$: $$(a+b)^{n+1} = (a+b)(a+b)^n = distribute = a^1b^0(a+b)^n +b^1a^0(a+b)^n$$

We see that step 3 will add 1 separately to each of the previously $a$ and $ b$ exponents and by induction we have our result like a domino game,

1. Since it is true for $n=1$ it must be true for $n+1=2$.
2. Since it is true for $n=2$ it must be true for $n+1=3$
3. And so on...

Answer 2

You multiply $n$ times $(a+b)$.

What things would you obtain?

By the rule of multiplying sums, any term of the product is obtained by taking either $a$ or $b$ from each bracket. Since there are $n$ brackets, you have to make $n$ choices. If you had chosen "$b$" $k$ times, you have $n-k$ times $a$ chosen.

Hence every term of the product is of the form $a^{n-k}b^k$ for some $k\in\{0,1,\dots,n\}$.