Question

determine the seventh term of the following expression without expanding the binomial.

\\(3x + 2y)^{16}\\

Explanation:

Identify the general term formula of the binomial expansion

$$ T_{r+1} = \binom{n}{r} a^{n-r} b^r $$

$$ (3x + 2y)^{16} \implies n = 16,\ a = 3x,\ b = 2y $$

Set the term index to find the seventh term

$$ T_7 \implies r + 1 = 7 \implies r = 6 $$

$$ T_7 = \binom{16}{6} (3x)^{16-6} (2y)^6 $$

Calculate the coefficient and simplify the expression

$$ \binom{16}{6} = \frac{16 \times 15 \times 14 \times 13 \times 12 \times 11}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 8008 $$

$$ (3x)^{10} = 59049 x^{10} $$

$$ (2y)^6 = 64 y^6 $$

$$ T_7 = 8008 \times 59049 \times 64 \times x^{10} y^6 = 30,263,024,640 x^{10} y^6 $$

Answer:

\(30,263,024,640 x^{10} y^6\)