18. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of
$$
(x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\cdots+(x+1)^{99} \text {. }
$$

Determine the value of $\left\lfloor a_{4} / a_{3}\right\rfloor$.
18. Answer: 19 .
Note that
$$
a_{k}=\binom{1}{k}+\binom{2}{k}+\binom{3}{k}+\binom{4}{k}+\cdots+\binom{99}{k}=\binom{100}{k+1}
$$

Thus
$$
a_{4} / a_{3}=\frac{\binom{100}{5}}{\binom{100}{4}}=\frac{96}{5}
$$

Thus the answer is 19 .