Question

Given six line segments of length $ 2,3,4,5,6,7 $ units. Then the number of triangles that can be formed by joining these lines is:
\[
  A.{\text{ }}{}^6{C_3} - 7 \\
  B.{\text{ }}{}^6{C_3} - 1 \\
  C.{\text{ }}{}^6{C_3} \\
  D.{\text{ }}{}^6{C_3} - 2 \\
 \]

Answer 1

Hint:In order to solve the problem and find the number of possible line segments forming circle first use the concept of permutation and combination to find out the total number of ways of selection of 3 lines segments from the given and further separate out those set which are practically not feasible to form a triangle using the properties related to length of sides of a triangle.

Complete step-by-step answer:
Given length of six line segments are:
 $ 2,3,4,5,6,7 $ units
Let us first use the concept of combination to find out the total number of possible selection of three different sides at once.
As we know that if we uniquely need to select $ y $ items from a set of $ x $ items, then the total number of selection is given by:
 $ {}^x{C_y} $
Using the above formula let us find the number of methods of selecting 3 line segments at once from the set of six line segments.
The number of ways is:
 $ {}^6{C_3} $
We have different sets but some of these will not be able to form a triangle on the basis of some limitations.
We know that the sum of lengths of any two sides of a triangle is always greater than the third side.
We have some set of length of line segments which does not fulfill this criterion so let us enlist those sets of lengths.
Such cases are:
 $
  2,3,5 \\
  2,3,6 \\
  2,3,7 \\
  2,4,6 \\
  2,4,7 \\
  2,5,7 \\
  3,4,7 \\
  $
We have a total of 7 such cases.
These 7 sets will not be able to form a triangle as they don't fulfill the basic criteria.
So let us subtract these terms.
Therefore the total counts of set that can form a triangle are:
 $ = {}^6{C_3} - 7 $
Hence, the number of triangles to that can be formed by joining these lines are $ {}^6{C_3} - 7 $
So, the correct answer is option A.

Note: In order to solve such problems students must remember the use of permutation and combination. Permutation is used when we need the number of different arrangements and combination is used when we need a number of different selections. Also students must remember to segregate out the number of those cases from the set which are not feasible to satisfy the given problem as we did in the above solution.