A bag contains 6 red, 5 white and 4 black balls. Two balls are drawn. Find the probability that none of them is red.

Answer

To find the probability that none of the two balls drawn from a bag containing 6 red, 5 white, and 4 black balls is red, we can follow these steps:

### Step 1: Calculate the Total Number of Balls
First, we need to find the total number of balls in the bag.

- Number of red balls = 6
- Number of white balls = 5
- Number of black balls = 4

**Total number of balls = 6 + 5 + 4 = 15**

### Step 2: Calculate the Total Ways to Draw 2 Balls
Next, we calculate the total number of ways to draw 2 balls from the 15 balls.

- The number of ways to choose 2 balls from 15 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose.

\[ \text{Total ways to choose 2 balls} = \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 \]

### Step 3: Calculate the Number of Favorable Outcomes (No Red Balls)
Now, we need to find the number of ways to draw 2 balls such that none of them is red.

- Since there are 6 red balls, the remaining balls (which are either white or black) are \( 5 + 4 = 9 \).

**The number of ways to choose 2 balls from these 9 (non-red) balls is:**

\[ \text{Ways to choose 2 non-red balls} = \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 4: Calculate the Probability
Finally, we can calculate the probability that none of the balls drawn is red.

\[ \text{Probability (none red)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{\binom{9}{2}}{\binom{15}{2}} = \frac{36}{105} \]

### Step 5: Simplify the Probability
Now, we simplify the fraction \( \frac{36}{105} \).

- Both 36 and 105 can be divided by 3:

\[ \frac{36 \div 3}{105 \div 3} = \frac{12}{35} \]

### Final Answer
Thus, the probability that none of the two balls drawn is red is:

\[ \boxed{\frac{12}{35}} \]