Question

Islamabad visited by 53 students, Lahore visited by 52 students, Murree visited by 65 students. Islamabad & Lahore visited by 19 students. Murree & Lahore visted by 28 studerts. Islamabad & Murree visited by 31

students. 38 of them visited exact'y one of the three cities like slamabad, Lohore & Murree. (15)

How many students visited only Islamabad?

How many students visited only Lahore?

How many stuadents visited only Murree?

How many students visited only Islamabad & Lahore but not Murree?

How many students Visited only Muree & Lahore but not Islamabad

How many students visited only Islamabad & Murree but not Lahore?

How many students visited all of the three cities?

Answer 1

Imagine the number of people who visited these cities in the form of Euler circles:

$X_1+X_5+X_2+X_3=53$

$X_5+X_4+X_3+X_6=52$

$X_2+X_3+X_6+X_7=65$

$X_5+X_3=19$

$X_3+X_6=28$

$X_2+X_3=31$

$X_1+X_4+X_7=38$ , therefore

$X_4+X_6=33$

$X_2+X_7=37$

$X_1+X_5=22$ , therefore

$X_1+X_5+X_2+X_7+X_4+X_6-$ $(X_1+X_4+X_7)=33+37+22-38$

$X_5+X_6+X_2=54$

We summarize the first three equations:

$X_1+X_5+X_2+X_3+ X_5$ $+X_4+X_3+X_6+ X_2+X_3+X_6+X_7$ $=53+52+65$

$(X_1+X_4+X_7)$ $+( 2\times X_5+2\times X_6+2\times X_2)$ $+3\times X_3=170$

$3\times X_3=170-108-38$

$X_3=8$ - students visited all three cities, therefore

$X_5=11$ -students visited only Islamabad & Lahore

$X_6=20$ -students Visited only Murree & Lahore

$X_2=23$ - students visited only Islamabad & Murree

$X_1=34-23=9$ - students visited only Islamabad

$X_4=33-20=13$ - students visited only Lahore

$X_7=34-20=14$ - students visited only Murree