Out of 32 persons, 30 invest in National Savings Certificates and 17 invest in shares. What is the number of persons who invest in both ?

Solution :
Let $$N = $$  National Savings Certificates
$$S = $$  Shares
Total number of persons $$ = 32$$
Number of persons who invest in National Savings Certificates $$ = 30$$
Number of persons who invest in Shares $$ = 17$$
Therefore
$$n\left( {N \cup S} \right) = 32,\,\,n\left( N \right) = 30,\,\,n\left( S \right) = 17$$
We know that,
$$\eqalign{ & n\left( {N \cup S} \right) = n\left( N \right) + n\left( S \right) - n\left( {N \cap S} \right) \cr & \Rightarrow 32 = 30 + 17 - n\left( {N \cap S} \right) \cr & \Rightarrow n\left( {N \cap S} \right) = 47 - 32 = 15 \cr} $$