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? Choose the correct choice from the following
A) 3  B) 4  C) 5  D) 6  E) 7
Explanation :   Let the lengths of the sides be a, b and c.   Given, Triagle's perimeter &lt; 14   =&gt; a+b+c &lt; 14 ----------(i)   From triangle Inequality theorem,   c &lt; a+b   =&gt; 2c &lt; a+b+c. ----------(ii)   From (i) and (ii), 2c &lt; a+b+c &lt; 14.   =&gt; c &lt; 7 ----------(iii)   =&gt; By similarity, we can say that length of any side must be less than 7.   Keeping these in mind, we can find the following possibilities :-   (2,3,4) (2,4,5) (2,5,6) (3,4,5) (3,4,6)   =&gt; Total 5 possibilities.   Answer : 5
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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?