State the binomial theorem for \((a+b)^n\), where \(n\) is a non-negative integer.

How many terms are in the expansion of \((x+y)^7\)?

Write the general term of \((a+b)^5\) using index \(k\).

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

Expand \((x+2)^3\).

Expand \((2x+y)^2\).

Find the fourth term in \((x+3)^6\), counting from the first term.

Find the coefficient of \(x^4\) in \((x+1)^6\).

Expand \((x-1)^4\).

Find the coefficient of \(x^3\) in \((2x-3)^5\).

Explain why the coefficients in \((x+y)^n\) are symmetric.

Use the binomial theorem to explain why the coefficients of \((x+y)^4\) are \(1,4,6,4,1\).

Find the constant term in \((2x+\frac1x)^6\).

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

Find all terms in \((x+2)^5\) that contain powers of \(x\) greater than \(2\).

Find the term independent of \(x\) in \((x^2-\frac{2}{x})^9\).

In \((a+b)^n\), determine which term contains \(a^3b^{n-3}\), and state its coefficient.

Diagnose the error: \((x+y)^3=x^3+y^3\).

Use the binomial theorem to prove \(\sum_{k=0}^{n}\binom nk=2^n\).

Find the coefficient of \(x^0\) in \((3x^2-\frac{1}{x})^{12}\), or explain why no such term exists.