Difficulty finding multinomial Coefficients

Question

Find the coefficient of $x^{18}$ in the expansion of $(1+x^{3}+x^{5}+x^{7})^{100}$

I got the following answer:

$\binom{100}{96,2,1,1} + \binom{100}{94,6,0,0}$

But the answer given is:

$\binom{100}{6} + 97 \times 98 \times \binom{100}{2} + 97 \times \binom{100}{3}$

Answer 1

Note that $$\begin{array}{} 18&=7+5+3*2&\equiv\binom{100}2\binom{98}1\binom{97}1&=\binom{100}2\times98\times97 \\&=5*3+3&\equiv\binom{100}3\binom{97}1&=\binom{100}3\times97 \\&=3*6&\equiv\binom{100}6&=\binom{100}6 \end{array}$$