Exercise
24
[205] Let \(F\) be a commutative ring, \(a,b∈ F\), \(n∈ℕ\) then
\[ (a+b)^{n}=∑ _{k=0}^{n}\binom {n}{k}a^{{n-k}}b^{k} \]
where the factor
\[ \binom n k {\stackrel{.}{=}}\frac{n!}{k!(n-k)!} \]
is called the ”binomial coefficient”. (This result is known as the binomial theorem, Newton’s formula, Newton’s binomial). To prove it by induction, check that
\[ \binom {n+1}{k+1}=\binom {n}{k+1}+\binom {n}{k} \]
for \(0≤ k,k+1≤ n\).