Вероятность разных цветов шаров

To solve this problem, we'll calculate the probability that both balls drawn are of different colors. The total number of balls and the number of ways to choose two balls of different colors will be essential in determining this probability.

Step 1: Calculate the Total Number of Balls

Total balls = Orange + Silver + Golden
Total balls = 18 + 29 + 21 = 68

Step 2: Calculate the Total Number of Ways to Choose Any Two Balls

The total number of ways to choose any two balls from 68 is given by the combination formula $C(n, k)$:

$$\text{Total ways} = C(68, 2) = \frac{68 \times 67}{2} = **2278**$$

Step 3: Calculate the Number of Ways to Choose Two Balls of Different Colors

We need to consider all possible pairs of different-colored balls:

1. Orange and Silver: $18 \times 29 = 522$ ways
2. Orange and Golden: $18 \times 21 = 378$ ways
3. Silver and Golden: $29 \times 21 = 609$ ways

Adding these up:

$$\text{Different color pairs} = 522 + 378 + 609 = **1509**$$

Step 4: Calculate the Probability

The probability $P$ that both balls are of different colors is:

$$P = \frac{\text{Number of ways to choose balls of different colors}}{\text{Total number of ways to choose any two balls}} = \frac{1509}{2278}$$

This fraction cannot be simplified further because the greatest common divisor (GCD) of 1509 and 2278 is 1.

Answer:

$\displaystyle \frac{1509}{2278}$