Quick questions on The binomial theorem and Pascal's triangle: expansion of $(a + b)^n$ and the general term

The $(k + 1)$-th term in the expansion of $(a + b)^n$ is

Each row of Pascal's triangle gives the coefficients of $(a + b)^n$ for that $n$. The triangle is built from two simple rules: every row begins and ends with $1$, and every interior entry is the sum of the two entries directly above it. That second rule is Pascal's rule, $\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$, so the triangle is a fast way to generate binomial coefficients for small $n$ without any factorials.

Coefficient of $x^m$ in $(a x + b)^n$: write $T_{k+1} = \binom{n}{k}(ax)^{n-k} b^k$, find the $k$ that gives $x^m$, then read the coefficient.

In $(a x^p + b x^q)^n$, the power of $x$ in $T_{k + 1}$ is $p(n - k) + q k$. Combine both contributions before setting the target; do not forget the power coming from the second term.

$(a - b)^n$ alternates signs because $(-b)^k$ contributes $(-1)^k$. Track the sign through to the final coefficient.