How many differently shaped triangles
exist in which no two sides are of the same length, each side is of integral
unit length and the perimeter of the triangle is less than 14 units?

A.3 B. 4 C. 5 D. 6 E. None of these

Solution follows here:

__Solution:__
Let
the lengths of the sides be a,b,c. Given that perimeter < 14 => a+b+c
< 14 ---(1)

From
triangle Inequality theorem, c < a+b => 2c < a+b+c

=>
From (1), 2c < a+b+c < 14 => c < 7 => By similarity, we can say
that length of any side must be less than 7.

Keeping
triangle inequality theorem, going by trial and error technique, we can find the following possibilities:

2,3,4; 2,4,5; 2,5,6; 3,4,5; 3,4,6;

=>
Total five possibilities.

**Answer (C)**
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