There are n points in a plane out of these points no three are in the same straight line except p points which are collinear. Let "k" be the number of straight lines and "m" be the number of triangles .Then find m−k ?(Assume n=7p=5)

Options

A. `""^(m+n)C_2`
B. `""^(n-m)C_2`
C. `""^(n)C_2-"^(p)C_ 2+1`
D. none of (A) ,(B ) ,(C)

Answer

To solve the problem, we need to find the difference between the number of triangles (m) and the number of straight lines (k) formed by n points in a plane, given that no three points are collinear except for p points which are collinear.

Given:
- \( n = 7 \)
- \( p = 5 \)

### Step 1: Calculate the number of straight lines (k)

The total number of straight lines formed by n points can be calculated using the combination formula for choosing 2 points from n points, which is given by:

\[ k = \binom{n}{2} - \binom{p}{2} + 1 \]

Where:
- \( \binom{n}{2} \) is the number of lines formed by any two points from n points.
- \( \binom{p}{2} \) accounts for the lines formed by the p collinear points, which only count as one line.

Substituting the values:

\[ k = \binom{7}{2} - \binom{5}{2} + 1 \]

Calculating \( \binom{7}{2} \) and \( \binom{5}{2} \):

\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]
\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

Now substituting these values back into the equation for k:

\[ k = 21 - 10 + 1 = 12 \]

### Step 2: Calculate the number of triangles (m)

The number of triangles formed by choosing any three points from n points is given by:

\[ m = \binom{n}{3} - \binom{p}{3} \]

Where:
- \( \binom{n}{3} \) is the number of triangles formed by any three points from n points.
- \( \binom{p}{3} \) accounts for the triangles formed by the p collinear points, which do not form any triangle.

Substituting the values:

\[ m = \binom{7}{3} - \binom{5}{3} \]

Calculating \( \binom{7}{3} \) and \( \binom{5}{3} \):

\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
\[ \binom{5}{3} = \frac{5 \times 4}{2 \times 1} = 10 \]

Now substituting these values back into the equation for m:

\[ m = 35 - 10 = 25 \]

### Step 3: Calculate m - k

Now we can find \( m - k \):

\[ m - k = 25 - 12 = 13 \]

### Final Answer

Thus, the value of \( m - k \) is:

\[ \boxed{13} \]