The number of scalene triangles having all sides of integral length greater than 1 and perimeter less than 13 is?

 Let, A,B,C are three side of a triangle than 

Condition 1 -  A+B+C <  13

Condition 2 -  A,B,C > 1

Condition 3. Sum of the smaller side must be greater than the largest side => A+B > C

as Perimeter is less than 13 than longer side must be less than 13/2= 6.5

Why?

A+B > C

A+B+C > 2C 

C<(A+B+C)/2 < semi perimeter.

now, replacing A,B,C with (6-P), (6-Q), (6-R)

so, (6-P)+(6-Q)+(6-R) < 13

P+Q+R > 5

so solution is 

1. 2,2,2

2. 2,2,3

3. 3,2,4

4. 3,3,3

5. 3,3,4

6. 3,3,5

7. 4,3,4

8. 4,3,5

9. 4,4,2

10. 4,4,4

11. 5,2,4

12. 5,2,5

there are twelve triangle,