In an urn, there are several colored balls, with equal numbers of each color. We add 14 balls which are all of the same new color, that is different from those in the urn. It is calculated that the probability of drawing , without replacement, two balls of the same color is constant. How many balls are there in the urn initially?

144

254 To solve this problem, we need to set up an equation based on the given information. Let's break down the problem step by step:

1. Let's assume there are initially 'x' balls of each color in the urn.
2. After adding 14 balls of a new color, the total number of balls in the urn becomes 'x + 14'.
3. Since we are drawing two balls without replacement, the probability of drawing two balls of the same color from the urn can be represented as (x/(x + 14)) * ((x-1)/(x + 13)).

Now, according to the given information, this probability is constant. So, let's set up the equation:

(x/(x + 14)) * ((x-1)/(x + 13)) = constant

4. To simplify the equation, let's remove the constant and cross-multiply:

(x^2 - x) = (x + 14) * (x + 13)

5. Expanding the right side:

x^2 - x = x^2 + 27x + 182

6. Cancelling out the x^2 terms:

- x = 27x + 182

7. Rearranging the equation:

28x = -182

8. Dividing both sides by 28:

x = -182/28

Since the number of balls cannot be negative, we discard this solution. Hence, there is no valid solution for this problem, and we cannot determine the initial number of balls in the urn.