@tanveer-malhotra As a+b+c+d+e+f < = 17, implies it can vary from (6,16)
To solve the inequality and make it into an equation, he considered a dummy variable g such that
a+b+c+d+e+f+g = 17. g can vary from 0 to 11.
In the above equation, a,b,c,d,e,f vary from 1 to 6 while g varies form 0 to 11.
To bring them on a common platform, we assume a,b,c,d,e,f having an initial value of 1, which makes the equation a+b+c+d+e+f+g = 11 where all start at 0.
total solutions for this is 11+7-1 C 7-1 which 17 C 6.
However, we have to subtract cases of a,b,c,d,e,f exceeding 6.
For that , take a has taken a value of 6,
so a+b+c+d+e+f+g = 11-6 = 5.
Invalid cases for a is 5+7-1 C 7-1 which is 11C6.
consider the invalid cases for all variables which is 11C6 * 6.
and now subtract it from the actual cases.
@zabeer thanks for your response. Ya agreed that both would be same, but my query is that why don't we add both the values, i.e, aren't the events of chosing smaller table first followed by the larger table, and chosing the larger table first followed by the smaller table, exhaustive events in themselves?
As both the events could be possible, and suit our requirement, so i believe that the total number of ways possible should be 2*12C5 x 4! x 6!
Number of Triangles with Integer sides for a given perimeter.
If the perimeter p is even then, total triangles is [p^2]/48.If the perimeter p is odd then, total triangles is [(p+3)^2]/48If it asks for number of scalene triangle with a given perimeter P, then subtract 6 and apply the same formula . For even [(p-6)^2]/48 and for odd [(p-3)^2]/48.
Where [x] represents neatest integer function. For example [6.7] is 7 not 6 because its nearest integer.