Out of 15 points in a plane, n points are in the same straight line. 445 triangles can be formed by joining these points. What is the value of n?a)3b)4c)5d)6Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Solution:

To solve this problem, we need to use the concept of combinations and counting principles.

Step 1: Counting the number of triangles formed by selecting any 3 points out of 15.

The number of ways to select 3 points out of 15 is given by the combination formula C(15, 3):

C(15, 3) = 15! / (3!(15-3)!) = 15! / (3!12!) = (15 * 14 * 13) / (3 * 2 * 1) = 455

So, there are a total of 455 triangles that can be formed by selecting any 3 points out of the 15.

Step 2: Counting the number of triangles formed by selecting 3 points in a straight line.

If n points are in the same straight line, then any triangle formed by selecting 3 points from these n points will be a straight line itself and hence not a valid triangle. So, we need to subtract the number of straight lines from the total number of triangles.

The number of straight lines that can be formed by selecting any 2 points out of n is given by the combination formula C(n, 2):

C(n, 2) = n! / (2!(n-2)!) = n! / (2! (n-2)!) = (n * (n-1)) / (2 * 1) = (n^2 - n) / 2

So, the number of triangles formed by selecting 3 points in a straight line is given by:

Number of triangles = C(n, 3) - C(n, 2) = (n! / (3!(n-3)!)) - ((n^2 - n) / 2)

We are given that the number of triangles formed by selecting any 3 points out of 15 is 445. So, we can write the equation as:

445 = C(15, 3) - ((n^2 - n) / 2)

Substituting the value of C(15, 3) = 455 in the equation, we get:

445 = 455 - ((n^2 - n) / 2)

Simplifying the equation, we have:

10 = (n^2 - n) / 2

Multiplying both sides of the equation by 2, we get:

20 = n^2 - n

Rearranging the equation, we have:

n^2 - n - 20 = 0

Factoring the quadratic equation, we have:

(n - 5)(n + 4) = 0

So, n = 5 or n = -4

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