In a plane, there are 16 non-collinear points. Find the number of straight lines formed.

Answer

To find the number of straight lines that can be formed by 16 non-collinear points in a plane, we can use the concept of combinations. Here’s the step-by-step solution:

### Step 1: Understanding the Problem
We need to find the number of straight lines that can be formed by joining any two points from a set of 16 non-collinear points. Since no three points are collinear, any two points will form a unique straight line.

### Step 2: Using Combinations
To determine how many ways we can choose 2 points from the 16 points, we use the combination formula:

\[ nCr = \frac{n!}{r!(n-r)!} \]

In this case, \(n = 16\) (the total number of points) and \(r = 2\) (the number of points we are choosing to form a line).

### Step 3: Applying the Combination Formula
Substituting the values into the formula gives us:

\[ 16C2 = \frac{16!}{2!(16-2)!} = \frac{16!}{2! \cdot 14!} \]

### Step 4: Simplifying the Factorials
Now, we can simplify the factorials:

\[ = \frac{16 \times 15 \times 14!}{2! \times 14!} \]

The \(14!\) cancels out:

\[ = \frac{16 \times 15}{2!} \]

### Step 5: Calculating \(2!\)
Calculating \(2!\):

\[ 2! = 2 \times 1 = 2 \]

### Step 6: Final Calculation
Now substituting back into our equation:

\[ = \frac{16 \times 15}{2} = \frac{240}{2} = 120 \]

### Conclusion
Thus, the total number of straight lines that can be formed by joining any two of the 16 non-collinear points is **120**.

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