Question

In a survey conducted on a group of 1000 people it is found that 40% buy product A, 20% buy product B, and 10% buy Product C, 5% buy products A and B, 3% buy products B and C and 4% buy products A and C. If 2% of the group buy all the three products, then find the number of people who buy i) product A only. ii) atleast one of the products A, B or C ​​

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{P},\mathrm{Q}\mathrm{and}\mathrm{R}\mathrm{denote}\mathrm{the}\mathrm{sets}\mathrm{of}\mathrm{people}\mathrm{buying}\mathrm{product}\mathrm{A},\mathrm{B},\mathrm{C} \\ \mathrm{respectively}.\mathrm{Let}\mathrm{U}\mathrm{be}\mathrm{the}\mathrm{universal}\mathrm{set}.\mathrm{Then} \\ \mathrm{n}\left(\mathrm{P}\right)=40\%\mathrm{of}1000=400 \\ \mathrm{n}\left(\mathrm{Q}\right)=20\%\mathrm{of}1000=200 \\ \mathrm{n}\left(\mathrm{R}\right)=10\%\mathrm{of}1000=100 \\ \mathrm{n}\left(\mathrm{P}\cap \mathrm{Q}\right)=5\%\mathrm{of}1000=50, \\ \mathrm{n}\left(\mathrm{Q}\cap \mathrm{R}\right)=3\%\mathrm{of}1000=30 \\ \mathrm{n}\left(\mathrm{R}\cap \mathrm{P}\right)=4\%\mathrm{of}1000=40 \\ \mathrm{n}\left(\mathrm{P}\cap \mathrm{Q}\cap \mathrm{R}\right)=2\%\mathrm{of}1000=20 \\ \mathrm{n}\left(\mathrm{U}\right)=1000 \\ \left(1\right)\mathrm{Required}\mathrm{number}=\mathrm{n}\left(P\cap \mathrm{Q}\text{'}\cap \mathrm{R}\text{'}\right)=\mathrm{n}\left(\mathrm{P}\cap \left(\mathrm{Q}\cup \mathrm{R}\right)\text{'}\right) \\ =\mathrm{n}\left(\mathrm{P}\right)-\mathrm{n}\left[\mathrm{P}\cap \left(\mathrm{Q}\cup \mathrm{R}\right)\right] \\ =\mathrm{n}\left(\mathrm{P}\right)-\mathrm{n}\left[\left(\mathrm{P}\cap \mathrm{Q}\right)\cup \left(\mathrm{P}\cap \mathrm{R}\right)\right] \\ =\mathrm{n}\left(\mathrm{P}\right)-\left[\mathrm{n}\left(\mathrm{P}\cap \mathrm{Q}\right)+\mathrm{n}\left(\mathrm{P}\cap \mathrm{R}\right)-\mathrm{n}\left\{\left(\mathrm{P}\cap \mathrm{Q}\cap \mathrm{R}\right)\right\}\right] \\ =400-\left(50+40-20\right)=330 \\ \left(2\right) \\ \mathrm{Atleast}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{products}\mathrm{A},\mathrm{B}\mathrm{or}\mathrm{C} \\ \mathrm{n}\left(\mathrm{P}\cup \mathrm{Q}\cup \mathrm{R}\right)=\mathrm{n}\left(\mathrm{P}\right)+\mathrm{n}\left(\mathrm{Q}\right)+\mathrm{n}\left(\mathrm{R}\right)-\mathrm{n}\left(\mathrm{P}\cap \mathrm{Q}\right)-\mathrm{n}\left(\mathrm{Q}\cap \mathrm{R}\right)-\mathrm{n}\left(\mathrm{R}\cap \mathrm{P}\right)+\mathrm{n}\left(\mathrm{P}\cap \mathrm{Q}\cap \mathrm{R}\right) \\ =400+200+100-50-30-40-20 \\ =700-140 \\ =560 \end{gathered}$$
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