If α represents the number of arrangements of p men and q women in a row such that all men are together and β represents the number of circular arrangements of the same people with the same condition, then α: β is

Question

If α represents the number of arrangements of p men and q women in a row such that all men are together and β represents the number of circular arrangements of the same people with the same condition, then α: β is (A) (q+1) p !: 1 (B) (q+1): 1 (C) 1: p ! (D) p !: q !

Tardigrade Admin · Accepted Answer

Given all men are together ∴ Linear arrangement of p men and q women =p !(q+1) ! and circular arrangement of p men and q women =p ! q ! ⇒ (α/β)=(p !(q+1) !/p ! q mid !)=(q+1): 1

Q. If $α$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $β$ represents the number of circular arrangements of the same people with the same condition, then $α : β$ is

$(q + 1) p! : 1$

$(q + 1) : 1$

$1 : p!$

$p! : q!$

Given all men are together
$∴$ Linear arrangement of $p$ men and $q$ women $= p! (q + 1)!$
and circular arrangement of $p$ men and $q$ women $= p! q!$
$\Rightarrow \frac{α}{β} = \frac{p ! ( q + 1 )!}{p ! q ∣ !} = (q + 1) : 1$

Q. If α represents the number of arrangements of p men and q women in a row such that all men are together and β represents the number of circular arrangements of the same people with the same condition, then α:β is

(q+1)p!:1

(q+1):1

1:p!

p!:q!