Question

determine the sixth term of the following expression without expanding the binomial. $(3x + 5y)^{12}$

Explanation:

Step1: Recall binomial term formula

The $k$-th term of $(a+b)^n$ is $\binom{n}{k-1}a^{n-(k-1)}b^{k-1}$. For 6th term, $k=6$, $n=12$, $a=3x$, $b=5y$.

Step2: Calculate binomial coefficient

$\binom{12}{6-1} = \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792$

Step3: Compute $a$ component

$a^{12-(6-1)} = (3x)^7 = 3^7x^7 = 2187x^7$

Step4: Compute $b$ component

$b^{6-1} = (5y)^5 = 5^5y^5 = 3125y^5$

Step5: Multiply all components

$792 \times 2187 \times 3125 \times x^7y^5$
First calculate the constants: $792 \times 2187 = 1,732,104$; $1,732,104 \times 3125 = 5,412,825,000$

Answer:

$5412825000x^7y^5$