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From Lohr's Sampling and Design Textbook

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My thoughts are that since in each stratum, we have $N/H$ units, and each at each stratum we want to have $n/H$ units from that stratum, then there are ${N/H \choose n/H} * H$ possible samples. But my thoughts are since this produces $\frac{N/H!}{n/H!(\frac{N-n}{H})!}*H$, this isn't the correct way to solve this question

Also, as a user commented, is this not valid since each stratum will have different proportions, and thus different $N/H$ and $n/H$ values, so we can not multiply it by H?

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    Since each stratum has different proportional allocation, i.e. each stratum has different sampling units, wouldn't multiplying the combination by H be incorrect? – 2017-02-13

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