Out of the four arbitrary non - collinear points three points are taken at a time, then the number of planes that can drawn through the three points is :

Options

A. 3
B. 4
C. 6
D. 12

Answer

To solve the problem of finding the number of planes that can be drawn through three points selected from four arbitrary non-collinear points, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 4 non-collinear points, and we need to determine how many different sets of 3 points can be selected from these 4 points. Each set of 3 points will define a unique plane.

2. **Using Combinations**: The number of ways to choose 3 points from 4 points can be calculated using the combination formula, which is denoted as \( C(n, r) \) or \( nCr \). The formula for combinations is:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points, and \( r \) is the number of points to choose.

3. **Substituting Values**: In our case, \( n = 4 \) and \( r = 3 \). Therefore, we need to calculate \( C(4, 3) \):
\[ C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} \]

4. **Calculating Factorials**: Now, we calculate the factorials:
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 1! = 1 \)

5. **Substituting Factorials into the Formula**:
\[ C(4, 3) = \frac{24}{6 \cdot 1} = \frac{24}{6} = 4 \]

6. **Conclusion**: Therefore, the number of planes that can be drawn through any three of the four points is **4**.

### Final Answer:
The number of planes that can be drawn through three points selected from four arbitrary non-collinear points is **4**.