/sci/ - Science & Math

>>15159206
I am trying to prove the following claim:
For [math]a, b, n \in \mathbb{Z}_{\ge 1}, \; a > b[/math]
[eqn] \sum_{i=1}^n \binom{n}{i} b^i {(a-b)}^{n-i} = a^n - {(a-b)}^n. [/eqn]
This claim is derived from two solutions I came up with for the following problem:
How many strings of letters (from the English alphabet) of length 8 with at least 1 vowel (vowels = {A, E, I, O, U}) exist?
The obvious solution to this is [math]26^n - 21^n[/math], but [math] \sum_{i=1}^n \binom{n}{i} 5^i \cdot 21^{n-i}[/math] seems to work as well.

Can anyone help prove my claim? I tried using weak and strong induction but ran in to dead ends (pic related).