A box contains 3 red, 4 white and 2 black balls. The number of ways in which 3 balls can be drawn from the box, so that at least one ball is red , is ( all balls are different )

Options

A. 45
B. 64
C. 84
D. 85

Answer

To solve the problem of finding the number of ways to draw 3 balls from a box containing 3 red, 4 white, and 2 black balls such that at least one ball is red, we can follow these steps:

### Step 1: Calculate the total number of balls
The box contains:
- 3 red balls
- 4 white balls
- 2 black balls

Total number of balls = 3 + 4 + 2 = 9 balls.

### Step 2: Calculate the total number of ways to choose 3 balls from 9
We can use the combination formula \( nCk \) which is given by:
\[ nCk = \frac{n!}{k!(n-k)!} \]
For our case, we need to calculate \( 9C3 \):
\[ 9C3 = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]
So, there are 84 ways to choose any 3 balls from the 9 balls.

### Step 3: Calculate the number of ways to choose 3 balls with no red balls
If we want to find the number of ways to choose 3 balls such that none of them is red, we can only choose from the 6 non-red balls (4 white + 2 black).

Now we calculate \( 6C3 \):
\[ 6C3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
So, there are 20 ways to choose 3 balls without any red balls.

### Step 4: Calculate the number of ways to choose 3 balls with at least one red ball
To find the number of ways to choose 3 balls such that at least one is red, we can subtract the number of ways to choose 3 balls with no red balls from the total number of ways to choose 3 balls.

Thus, the number of ways to choose 3 balls with at least one red ball is:
\[ \text{Ways with at least one red} = \text{Total ways} - \text{Ways with no red} \]
\[ = 84 - 20 = 64 \]

### Final Answer
The number of ways to draw 3 balls from the box such that at least one ball is red is **64**.

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