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If $\alpha$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $\beta$ represents the number of circular arrangements of the same people with the same condition, then $\alpha: \beta$ is
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The correct answer is:
$(q+1): 1$
Given all men are together
$\therefore$ Linear arrangement of $p$ men and $q$ women $=p !(q+1) !$
and circular arrangement of $p$ men and $q$ women $=p ! q !$
$\Rightarrow \quad \frac{\alpha}{\beta}=\frac{p !(q+1) !}{p ! q \mid !}=(q+1): 1$
$\therefore$ Linear arrangement of $p$ men and $q$ women $=p !(q+1) !$
and circular arrangement of $p$ men and $q$ women $=p ! q !$
$\Rightarrow \quad \frac{\alpha}{\beta}=\frac{p !(q+1) !}{p ! q \mid !}=(q+1): 1$
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