NCERT Class XI Mathematics - Sets - Solutions
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Question : 73
Total: 73
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.
Solution:
Here n(A) = a + b + c + d = 21 ... (i)
n(B) = b + c + f + g = 26 ... (ii)
n(C) = c + d + e + f = 29 ... (iii)
n(A ∩ B) = b + c = 14 ... (iv)
n(C ∩ A) = c + d = 12 ... (v)
n(B ∩ C) = c + f = 14 ... (vi)
n(A ∩ B ∩ C) = c = 8 ... (vii)
Putting value of c in (iv), (v) and (vi),
b + 8 = 14 ⇒ b = 6, 8 + d = 12 ⇒ d = 4, 8 + f = 14 ⇒ f = 6
Putting value of c, d, f in (iii), 8 + 4 + e + 6 = 29
⇒ e = 29 – 18 = 11
Number of people who like product C only = 11.
n(B) = b + c + f + g = 26 ... (ii)
n(C) = c + d + e + f = 29 ... (iii)
n(A ∩ B) = b + c = 14 ... (iv)
n(C ∩ A) = c + d = 12 ... (v)
n(B ∩ C) = c + f = 14 ... (vi)
n(A ∩ B ∩ C) = c = 8 ... (vii)
Putting value of c in (iv), (v) and (vi),
b + 8 = 14 ⇒ b = 6, 8 + d = 12 ⇒ d = 4, 8 + f = 14 ⇒ f = 6
Putting value of c, d, f in (iii), 8 + 4 + e + 6 = 29
⇒ e = 29 – 18 = 11
Number of people who like product C only = 11.
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