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In a Survey It Was Found that 21 Persons Liked Product P1, 26 Liked Product P2 and 29 Liked Product P3. If 14 Persons Liked Products P1 and P2; - Mathematics

In a survey it was found that 21 persons liked product P1, 26 liked product P2 and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked product P3 and P1 ; 14 persons liked products P2 and P3 and 8 liked all the three products. Find how many liked product P3 only.

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Let \[P_1 , P_2 \text{ and } P_3\] denote the sets of persons liking products\[P_1 , P_2 \text{ and } P_3\] respectively.
Also, let U be the universal set.
Thus, we have:

n( \[P_1\]= 21, n(\[P_2\]= 26 and n(\[P_3\] 29
n(\[P_1\]\[\cap\]\[P_3\]= 12, n(\[P_2 \cap P_3\]n(\[P_1 \cap P_2 \cap P_3\]= 8 

Number of people who like only product \[P_3\] 

= `n (P_3∩ P_1′∩ P_2′)` 

= `n {P_3∩ (P_1∪  P_2)′}`

\[ = n \left( P_3 \right) - n\left[ P_3 \cap \left( P_1 \cup P_2 \right) \right]\]

\[ = n\left( P_3 \right) - n\left[ \left( P_3 \cap P_1 \right) \cup \left( P_3 \cap P_2 \right) \right]\]

\[ = n\left( P_3 \right) - \left[ n\left( P_3 \cap P_1 \right) + n\left( P_3 \cap P_2 \right) - n\left( P_1 \cap P_2 \cap P_3 \right) \right]\]

\[ = 29 - \left( 12 + 14 - 8 \right)\]

\[ = 11\]

Therefore, the number of people who like only product \[P_3\]is 11 

Concept: Universal Set
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RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.8 | Q 15 | Page 47
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