A bag contains 3 red, 7 white, and 4 black balls. If three balls are drawn from the bag, then find the probability that all of them are of the same color.

Explanation

To solve the problem of finding the probability that all three balls drawn from the bag are of the same color, we will follow these steps:

### Step 1: Determine the total number of balls in the bag.
The bag contains:
- 3 red balls
- 7 white balls
- 4 black balls

**Total number of balls = 3 + 7 + 4 = 14**

### Step 2: Calculate the total number of ways to select 3 balls from 14.
We can use the combination formula \( nCr = \frac{n!}{r!(n-r)!} \) to find the total ways to choose 3 balls from 14.

\[ \text{Total ways} = \binom{14}{3} = \frac{14!}{3!(14-3)!} = \frac{14!}{3! \cdot 11!} \]

Calculating this gives:

\[ \binom{14}{3} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = \frac{2184}{6} = 364 \]

### Step 3: Calculate the favorable outcomes for drawing 3 balls of the same color.
We need to consider three cases:
1. All 3 balls are red.
2. All 3 balls are white.
3. All 3 balls are black.

**Case 1: All 3 balls are red.**
\[ \text{Ways} = \binom{3}{3} = 1 \]

**Case 2: All 3 balls are white.**
\[ \text{Ways} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

**Case 3: All 3 balls are black.**
\[ \text{Ways} = \binom{4}{3} = 4 \]

### Step 4: Sum the favorable outcomes.
\[ \text{Total favorable outcomes} = 1 + 35 + 4 = 40 \]

### Step 5: Calculate the probability.
The probability \( P \) that all three balls drawn are of the same color is given by the ratio of favorable outcomes to total outcomes.

\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{40}{364} \]

### Step 6: Simplify the probability.
To simplify \( \frac{40}{364} \):

\[ P = \frac{10}{91} \]

### Final Answer:
The probability that all three balls drawn are of the same color is \( \frac{10}{91} \).

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