A market research group conducted a survey of $2, 000$ consumers and reported that $1720$ consumers liked product $P_{1}$ and $1, 4500$ consumers liked product $P_{2}$ . What is the least number that must have liked both the products.

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Solution

Total consumer = 2000
P1 liked by = 1720
P2 liked by = 1450
$∴ N (P_{1} \cup P_{2}) = N (P_{1}) + N (P_{2}) - N (P_{1} \cap P_{2})$
$\Rightarrow 2000 = 1720 + 1450 - N (P_{1} \cap P_{2})$
$\Rightarrow N (P_{1} \cap P_{2}) = 3170 - 2000$
$\Rightarrow 1170$
$∴$ least no : of people = 1170 liked both products.

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A market research group conducted a survey of 2,000 consumers and reported that 1720 consumers liked product P1 and 1,4500 consumers liked product P2. What is the least number that must have liked both the products.

Total consumer = 2000 P1 liked by = 1720 P2 liked by = 1450 ∴N(P1∪P2)=N(P1)+N(P2)−N(P1∩P2)⇒2000=1720+1450−N(P1∩P2)⇒N(P1∩P2)=3170−2000⇒1170∴ least no : of people = 1170 liked both products.

A market research group conducted a survey of $2, 000$ consumers and reported that $1720$ consumers liked product $P_{1}$ and $1, 4500$ consumers liked product $P_{2}$ . What is the least number that must have liked both the products.