Question

determine the seventh term of the following expression without expanding the binomial. $(x + y)^{16}$

Explanation:

Step1: Recall binomial term formula

The $k$-th term of $(a+b)^n$ is $\binom{n}{k-1}a^{n-(k-1)}b^{k-1}$

Step2: Identify values for the formula

Here, $n=16$, $k=7$, $a=x$, $b=y$. Calculate the exponent for $a$: $16-(7-1)=10$, exponent for $b$: $7-1=6$. Calculate the binomial coefficient: $\binom{16}{6}=\frac{16!}{6!(16-6)!}$

Step3: Compute the binomial coefficient

$\binom{16}{6}=\frac{16\times15\times14\times13\times12\times11}{6\times5\times4\times3\times2\times1}=8008$

Step4: Assemble the seventh term

Combine the coefficient and variable terms: $8008x^{10}y^6$

Answer:

$8008x^{10}y^6$