Binomial Theorem and Combinatorics (HSC Maths Extension 1, 2026) quiz

What is the general term $T_{k+1}$ in the expansion of $(a + b)^n$?

Evaluate the binomial coefficient $\binom{7}{4}$.

Find the coefficient of $x^3$ in the expansion of $(2x + 5)^7$.

Find the term independent of $x$ in the expansion of $\left(x^2 + \frac{3}{x}\right)^6$.

Find the term independent of $x$ in the expansion of $\left(2x - \frac{1}{x^2}\right)^9$.

What is the value of the sum $\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n}$?

Using $(1 + x)^n$ with $x = 2$, what does the sum $\binom{n}{0} + \binom{n}{1} \cdot 2 + \binom{n}{2} \cdot 2^2 + \cdots + \binom{n}{n} \cdot 2^n$ equal?

Find the coefficient of $x^2$ in the expansion of $(1 + x)^5 (1 + 2x)^4$.

For which value of $r$ is the binomial coefficient $\binom{14}{r}$ greatest?

In how many ways can 4 distinct letters be selected from the word MATHEMATICS so that the selection includes at least one vowel? Use the 8 distinct letters available.