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Each of $74$ students in a class studies at least one of the three subjects $H, E$ and $P$. Ten students study all three subjects, while twenty study $H$ and $E$, but not $P$. Every student who studies $P$ also studies $H$ or $E$ or both. If the number of students studying $H$ equals that studying $E$, then the number of students studying $H$ is _________

Given that,

• $n(H \cup E \cup P) = 74$
• $n( H \cap E \cap P) = 10$
• $n(H \cap E) = 20$
• Only $P = 0$

We can draw the Venn diagram,

Here, given that $: b=20, g=10, d=0$

The number of students studying $H =$ The number of students studying $E$

$a + b + e + g = b + c + g + f$

$\Rightarrow a + 20 + e + 10 = 20 + c + 10 + f$

$\Rightarrow \boxed{a + e = c + f} \quad \longrightarrow (1)$

We have, $a + b + c + d + e + f + g = 74$

$\Rightarrow a + 20 + c + 0 + e + f + 10 = 74$

$\Rightarrow a + e + c + f = 44$

$\Rightarrow 2(a+e) = 44 \quad [ \because \text{From equation (1)}]$

$\Rightarrow \boxed{a + e = 22}$

$\therefore$ The number of students who studies $H = a + b + e + g = 22 + 20 + 10 = 52.$

Correct Answer $: 52$

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