Ah, I understand what you mean now.
So the whole x(x-15) + x + (x-2) + 32 = 120 was:
x(x-15) is P(T) - the intersection of T and B (T n B)
x is the intersection of T and B (T n B)
and x-2 is P(B) - the intersection of T and B (T n B)
32 is the number of coins that are not from the 20th century AND are not British.
You know the total number of coins is 120 so all of the numbers in the circle and outside of the circle must add up to a total of 120. If the circles and the 32 outside of the circles had been fractions, the probabilities would add up to 1.
By equating the x equation with 32 to 120, you work back to find x. Obviously x>0 as you cannot have a negative number of coins.
I hope you understand now