Question

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Among 250 viewers interviewed who watch at least one of the three TV channels namely A, B &C. 116 watch A, 127 watch C, while 107 watch B. If 50 watch exactly two channels. How many watch exactly one channel?



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Answer 1

250 = n(Exactly 1 channel) + n(Exactly 2 channels) + n(Exactly 3 channels)

250 = n(Exactly 1 channel) + 50 + n(Exactly 3 channels)

Let's find the value of n(Exactly 3 channels) = x

250 = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C)

Note that each of n(A and B) is the sum of 'number of people watching exactly two channels A and B' and 'number of people watching all three channels'.

250 = 116 + 127 + 107 - n(Exactly 2 channels) - 3x + x

250 = 116 + 127 + 107 - 50 - 2x

x = 25

250 = n(Exactly 1 channel) + 50 + 25

n(Exactly 1 channel) = 175

Answer 175

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Answer 2

SOLUTION

GIVEN THAT

• n(At least one channel) = 250
• n(Exactly two channels) = 50

We know that

• n(At least one channel) = n(Exactly 1 channel) + n(Exactly 2 channels) + n(Exactly 3 channels) = 250
• 250 = n(Exactly 1 channel) + 50 + n(Exactly 3 channels)

Let’s find the value of n(Exactly 3 channels) = x

We know that

• n(A) + n(B) + n(C) – n(A and B) – n(B and C) – n(C and A) + n(A and B and C) = 250
• n(Exactly two channels) = n(A and B) + n(B and C) + n(C and A) – 3*n(A and B and C)
• n(A and B) + n(B and C) + n(C and A) = n(Exactly two channels) + 3*n(A and B and C)

Plug this into the equation above:

250 = n(A) + n(B) + n(C) – n(Exactly 2 channels) – 3*x+x

250 = 116 + 127 + 107 – 50 – 2x

x = 25

250 = n(Exactly 1 channel) + 50 + 25

n(Exactly 1 channel) = 175