Ravshonbek wrote:

There are 68 children in the cafeteria of a school and all of the children have something for lunch. Thirty-four of the children brought lunches from home, 23 of the children bought a drink from the cafeteria beverage machine, and 32 of the children bought fruit in the cafeteria. If 18 children did at least 2 of these things, how many children did exactly two of these things?

(A) 3

(B) 6

(C) 9

(D) 13

(E) 15

A - brought lunch from home, P(A) = 34

B - bought a drink, P(B) = 23

C - bought fruit, P(C) = 32

P(A) = P(A1)+P(A2)+P(A3) -- A1 those who only brought lunch from home, A2 - those who brought lunch from home and bought one item, A3 - brought lunch from home and bought two items

P(B) = P(B1)+P(B2)+P(B3)

P(C) = P(C1)+P(C2)+P(C3)

when we add them together P(A2) + P(B2) + P(C2) = 2*P2 //where P2 is the number of kids who did exactly two things, each of those kids is counted twice because if a kid brought lunch and bought a drink he belongs to A2 and B2. Consequenty, P(A3)=P(B3)=P(C3) = P3

P(A)+P(B)+P(C) = P1 + 2*P2+3*P3 = P(A)+P(B)+P(C) = 34+23+32 = 89

P2+P3 = 18 -> since 18 kids did at least two things

P1+P2+P3 = 68 -> because all 68 kids ate

P1+2*P2+3*P3 = 89

if we subtract second equation from the third one we get P2 + 2*P3 = 21, then multiply the first one by 2 and subtract the second one P2 = 18*2-21=36-21=15

E is the answer.