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How many ways are there to distribute $100$ different balls into $5$ different boxes so the last $2$ boxes contains even number of balls and first $3$ boxes contains balls more than $5$ and less than $20$ ?

For last $2$ boxes, I can apply the series $\left(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots\right)^2$

For the first three boxes, I am using the series as $\left(\frac{x^6}{6!}+\frac{x^7}{7!}+...... +\frac{x^{19}}{19!}\right)^3$

Am I right here or missing anything ?

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    Are you wanting exponential generating functions or ordinary generating functions... if exponential generating, then what might the part $(\frac{x^a}{a!}+\frac{x^{a+1}}{(a+1)!}+\dots+\frac{x^b}{b!})$ represent? How might you pick $a$ and $b$ to be useful here for you? – 2017-01-16
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    @JMoravitz I've edited. Can you please check now ? – 2017-01-16

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