Questions: Find the specified nth term in the expansion of the binomial. (x+y)^12, n=4

Solution Steps

We are tasked with finding the 4th term in the expansion of \((x+y)^{12}\). According to the binomial theorem, the \( k \)th term in the expansion is given by: \[ T_k = \binom{n}{k} x^{n-k} y^k \] Here, \( n = 12 \) and for the 4th term, \( k = 3 \).

We calculate the binomial coefficient \(\binom{12}{3}\): \[ \binom{12}{3} = 220 \]

For the 4th term, we find the powers of \( x \) and \( y \): \[ x^{12-3} = x^9 \quad \text{and} \quad y^3 = y^3 \]

Now we can express the 4th term: \[ T_4 = \binom{12}{3} x^{9} y^{3} = 220 x^{9} y^{3} \]

Final Answer