Final answer:
The coefficient of a³b² in the expansion of (3a + b)⁵ is found using the binomial theorem, leading to the coefficient of 270 for this specific term.
Step-by-step explanation:
To find the coefficient of a³b² in the expansion of (3a + b)⁵, we can use the binomial theorem. The general term in a binomial expansion (a + b)⁵ is given by T(k+1) = ⁵Ck a⁵-k b⁵k, where ⁵Ck are the binomial coefficients.
For the term containing a³b², we need the term where a is raised to the power of 3 and b raised to the power of 2. This corresponds to T(3) = ⁵C2 (3a)⁵-2 b², because k must be 2 to have b squared.
Calculating the binomial coefficient ⁵C2 = 10 and then substituting for ⁵C2 and the powers of a and b, we get:
T(3) = 10 × (3a)³ × b² = 10 × 27a³ × b² = 270a³b²
Therefore, the coefficient of the term a³b² in the expansion is 270.