Question

The number of noncongruent integer-sided triangles whose sides belong to the set $\{10,11,12,\dots .,22\}$ is

• A. $283$
• B. $446$
• C. $448$
• D. $449$

Answer 1

The correct option is C $448$

Number of scalene triangles
${=}^{13}{C}_3-3=283$
Combination which cannot be a scalene triangles $(10,11,22),(10,12,22),(10,11,21)$

Number of isosceles triangles
$={(}^{13}{C}_2\times 2)-4=152$
Combination which cannot be a isosceles triangles $(10,10,22),(11,11,22),(10,10,21),(10,10,20)$

Number of equilateral triangles
${=}^{13}{C}_1=13$

Total number of triangles $=448$