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CMI-2019-DataScience-A: 8

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Consider the following Venn diagram. The universal set $U$ is the set of all natural numbers from $1$ to $1000.$ 

The sets $A,B,C$ contain integers in $U$ that are multiples of $6,7,8$ respectively. The number of elements in the shaded region is:

  1. $12$
  2. $15$
  3. $16$
  4. $17$
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Here we need to find the difference between the number of elements in the sets  $B \bigcap C - A\bigcap B\bigcap C$ because we need to find only those numbers which are the multiples of 7 and 8 but not of 6. Now,

$\left | B\bigcap C \right | = \left \lfloor \frac{1000}{LCM (7,8)} \right \rfloor = \left \lfloor \frac{1000}{56} \right \rfloor = $ 17

$\left | A \bigcap B\bigcap C \right | = \left \lfloor \frac{1000}{LCM (6,7,8)} \right \rfloor = \left \lfloor \frac{1000}{168} \right \rfloor =$ 5

Hence elements in the shaded region = 12
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