Probability theory in card

Question

I trying to solve it exercise using excel, but it's too hard We have a deck of 11 cards. There are 2 black cards in the deck, 112 white and green cards more than yellow ones.
A = 11 B = 122 C = X2 D = Y.
You draw two random cards from the deck and have a 27% chance of drawing cards with the same color. How many green cards? Thanks!

Answer 1

Let a deck with $22$ black cards, $18$ white cards, $x$ green cards and $60-x$ yellow ones. Your condition is $x>60-x$, i.e. $x>30$.

The probability of $A:$ "draw two cards with the same color" is, $$\mathbf{P}(A)=\frac{22}{100}\frac{21}{99}+\frac{18}{100}\frac{17}{99}+\frac{x}{100}\frac{x-1}{99}+\frac{60-x}{100}\frac{59-x}{99}=\frac{2x^2-120x+4308}{9900}=\frac{27}{100}$$ The parabola is solved with $2x^2-120x+1635=0$, i.e. $x=39.083$ or $x=20.91$ (they are not integers). If your assume $x=39$, the exactly probability is: $$\mathbf{P}(A)=\frac{22}{100}\frac{21}{99}+\frac{18}{100}\frac{17}{99}+\frac{39}{100}\frac{38}{99}+\frac{21}{100}\frac{20}{99}=0.2718$$

With exactly $27\%$ there is no solution. But with $27.18\%$, the solution is $39$ green cards.