Fourth Term Expansion 500B4F

1. **State the problem:** We need to find the fourth term in the expansion of the binomial expression $$ (3a + b)^7 $$. 2. **Recall the binomial theorem formula:** The expansion of $$ (x + y)^n $$ is given by $$ \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k $$ where $$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$ is the binomial coefficient. 3. **Identify the terms:** Here, $$ x = 3a $$, $$ y = b $$, and $$ n = 7 $$. The $$ (r+1)^{th} $$ term in the expansion is $$ T_{r+1} = \binom{7}{r} (3a)^{7-r} b^r $$ for $$ r = 0,1,2,...,7 $$. 4. **Find the fourth term:** The fourth term corresponds to $$ r = 3 $$ (since the first term is $$ r=0 $$). So, $$$ T_4 = \binom{7}{3} (3a)^{7-3} b^3 = \binom{7}{3} (3a)^4 b^3 $$$ 5. **Calculate the binomial coefficient:** $$ \binom{7}{3} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 $$ 6. **Calculate the powers:** $$ (3a)^4 = 3^4 a^4 = 81 a^4 $$ 7. **Combine all parts:** $$ T_4 = 35 \times 81 a^4 \times b^3 = 2835 a^4 b^3 $$ **Final answer:** The fourth term of the expansion is $$ \boxed{2835 a^4 b^3} $$.