Question

What is there to know about the Binomial Coefficients and the Binomial Theorem?

Answer 1

When a binomial is raised to whole number powers, the coefficients of the terms in the expansion form a pattern.
\[
\begin{array}{cc}
(a+b)^{0}= & 1 \\
(a+b)^{1}= & 1 a+1 b \\
(a+b)^{2}= & 1 a^{2}+2 a b+1 b^{2} \\
(a+b)^{3}= & 1 a^{3}+3 a^{2} b+3 a b^{2}+1 b^{3} \\
(a+b)^{4}= & 1 a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+1 b^{4} \\
(a+b)^{5}= & 1 a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+1 b^{5}
\end{array}
\]