A. 120
Room1: 2a+1
Room2: 2b+1
Room3: 2c+1
(2a+1) + (2b+1) + (2c+1) = 31
a+b+c = 14.
whose solution is C(14+3-1,3-1) = C(16,2) = 120
B. Don't know
C. 0
number of solutions x1+x2+x3=20 (Solve it using Multinomial Theorem)
is coefficient of x^20 in (x^3+x^4+x^5)^2 * (x+x^2+..x^4)
i.e coefficient of x^13 in (1+x+x^2)^2 * (1+x+...x^3)
Since highest power in the above equation is x^7. Therefore coefficient of x^13 will be zero.