Q42 / 90 · KVPY 2016 (SB/SX) · Mathematics
The number of noncongruent integer-sided triangles whose sides belong to the set $\{10,11,12, \ldots ., 22\}$ is-
- A283
- B446
- Correct answerC448
- D449
Solution
Number of scalene triangles
$\begin{array}{l}
={ }^{13} \mathrm{C}_{3}-3 \\
=283
\end{array}\left\{\begin{array}{l}
10,11,22 \\
10,12,22 \\
10,11,21
\end{array}\right\}$
Number of isosceles triangles
$=\left({ }^{13} \mathrm{C}_{2} \times 2\right)-4 \quad\left\{\begin{array}{l}10,10,22 \\ 11,11,22 \\ =152\end{array}\right.$
Number of equilateral triangles
$={ }^{13} \mathrm{C}_{1}=13$
So total number of triangles $=448$
Hence, the correct answer is option: 'C'.