In a locality of 40 houses, each house is subscribed to at least one

Question

In a locality of 40 houses, each house is subscribed to at least one newspaper from amongst, The Sun, Daily Mail and Daily Mirror. 24 houses subscribe to Daily Mirror. 13 houses subscribe to both The Sun and Daily Mirror. 26 houses subscribe to Daily Mail out of which 16 also subscribe to The Sun. There is no house which subscribes to The Sun alone. 7 houses subscribe to all three newspapers. How many houses subscribe to only two newspapers?

(A) 12
(B) 14
(C) 16
(D) 18
(E) 20

Show Answer

Official Answer

D

(A) 12
(B) 14
(C) 16
(D) 18
(E) 20

IanStewart · Accepted Answer

7 houses get all three papers.

13 get the Sun and the Mirror (and possibly the Mail), so 6 houses get  only  the Sun and the Mirror.
16 get the Sun and the Mail (and possibly the Mirror), so 9 houses get  only  the Sun and the Mail.

Finally, 26 get the Mail. The rest, or 14 houses, must get the Sun only, the Sun and the Mirror only, or the Mirror only. But we know 0 get the Sun only, and 6 get the Sun and the Mirror only. The remaining 8 must get the Mirror only. Since 24 houses in total get the Mirror, and since 7 get all three, 6 get the Sun and Mirror only, and 8 get the Mirror only, the remaining 3 houses must get only the Mirror and the Mail.

So 6 + 9 + 3 = 18 houses get exactly two papers.

Of course it's easier to draw a Venn diagram when working all of this out.

Shivya290497 · Answer

Can someone post the solution please?

Aryaa03 · Answer

Option-D 
Sol: Solution using the overlapping sets formula (if nothing else works out for you)

Designate Sun as "S", Daily Mail as "D" and Daily Mirror as "M"

Total House = 40 ---> P(S u D u M) = 40
Each House has at least 1 ---> P(Neither)=0
24 houses subscribe to Daily Mirror ----> P(M)=24
13 houses subscribe to both The Sun and Daily Mirror ----> P(M n S) = 13
26 houses subscribe to Daily Mail out of which 16 also subscribe to The Sun -----> P(D) = 26 and P(S n D) = 16
7 houses subscribe to all three newspapers - P (S n D n M) = 7
There is no house which subscribes to The Sun alone: If no house subscribes to The Sun alone, P(S) = P(S n D) + P (M n S) - P (S n D n M) = 13+16-7 = 22
(The intersection of the three is subtracted because it is counted twice)

Now, 
P(S u D u M) = P(S) + P(D) + P(M) - P(S n D) - P(D n M) - P(M n S) + P (S n D n M)

From above, P(D n M) can be calculated as 10.

Only 2 are given as : P(S n D) + P(D n M) + P(M n S) - 3*P (S n D n M) = 10 + 13 + 16 - 3*7 = 18

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In a locality of 40 houses, each house is subscribed to at least one

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