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There is a class of $120$ students in St. Pauľs College. All the students are numbered $1$ to $120$, where in all even numbered students opt for History, those numbers are divisible by $5$ opt for Geography and those whose numbers are divisible by $7$ opt for Political Science. How many opt for none of the three subjects ? 

  1. $19$
  2. $41$
  3. $21$
  4. $57$
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Option B. 41

Here n(H) = $\frac{120}{2}$ = 60, n(G) = $\frac{120}{5}$ = 24, n(PS) = $\frac{120}{7}$ = 17            [ taking quotient only]

n($H\cap G$) = $\frac{120}{10}$ = 12, n($H\cap PS$) = $\frac{120}{14}$ = 8, n($G\cap PS$) = $\frac{120}{35}$ = 3

n($H\cap G\cap PS$) = $\frac{120}{70}$ = 1

Using Venn Diagrams

Number of student opt for any of three subjects = n( H ∪ G ∪ PS ) = n(H) + n(G) + n( PS ) – n( H ∩ G ) – n ( G ∩ PS ) – n ( PS ∩ H ) + n (H ∩ G ∩ PS)

= 60 + 24 + 17 – 12 – 8 – 3 + 1 = 79

Hence, Number of student opt for none of three subjects = 120 – 79 = 41

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