The number of non-congruent integer-sided tri | vedclass

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(C) Let the set of side lengths be $S = \{10, 11, \ldots, 22\}$. The number of elements in $S$ is $n = 13$.For a triangle with sides $a, b, c \in S$,t

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$448$

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(C) Let the set of side lengths be $S = \{10, 11, \ldots, 22\}$. The number of elements in $S$ is $n = 13$.For a triangle with sides $a, b, c \in S$,the triangle inequality requires $a+b > c$,$a+c > b$,and $b+c > a$. Assuming $a \le b \le c$,this is equivalent to $a+b > c$.Total combinations of sides $(a, b, c)$ with $a \le b \le c$ is given by the stars and bars approach or direct counting.$1$. Equilateral triangles $(a=b=c)$: There are $13$ choices.$2$. Isosceles triangles ($a=b 
eq c$ or $a=c 
eq b$ or $b=c 
eq a$): For $a=b$,we need $2a > c$. For each $a \in \{10, \ldots, 22\}$,$c$ can be any value in $S$ such that $c $3$. Scalene triangles $(a  c$. The total number of such triangles is $283$.Total number of triangles $= 283 + 152 + 13 = 448$.

The number of non-congruent integer-sided triangles whose sides belong to the set $\{10, 11, 12, \ldots, 22\}$ is

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