Binomial expansion: แบบฝึกหัด

Find the coefficient of \(x^2\) in the expansion of \((3+x)^4\).

Find the sum of the coefficients in the expansion of \( (2x-1)^5 \).

Find the coefficient of \(x^3\) in the expansion of \( (1+2x)(1+x)^5 \).

Find the coefficient of \(x^5\) in the binomial expansion of \( (1 + 4x)^7 \).

Given that the coefficient of \(x\) in the expansion of \((1-ax)(1+x)^5\) is \(10\), find the value of the constant \(a\).

Determine the number of terms in the expansion of \((a+b)^{2024}\).

If the sum of all binomial coefficients in the expansion of $$(1+x)^n$$ is $$256$$, find the coefficient of $$x^3$$.

Given that the coefficient of $$x^2$$ in the expansion of $$(1+3x)^n$$ is $$324$$, where $$n$$ is a positive integer. Find the coefficient of $$x^3$$ in the expansion of $$(1-2x)(1+3x)^n$$.

Find the coefficient of \(x\) in the expansion of \((3x - \frac{2}{x})^5\).

Consider the binomial expansion of \((x^2 + \frac{k}{x})^6\), where \(k\) is a positive constant.
(a) Find the general term in the expansion.
(b) If the term independent of \(x\) in the expansion is 240, find the value of \(k\).
(c) Using the value of \(k\) found in (b), find the coefficient of \(x^3\) in the expansion.