Let $${n_1}\, < {n_2}\, < \,{n_3}\, < \,{n_4}\, < {n_5}$$ be positive integers such that $${n_1}\, + {n_2}\, + \,{n_3}\, + \,{n_4}\, + {n_5}$$ = 20. Then the number of such destinct arrangements $$\,({n_1}\,,\,{n_2},\,\,{n_3},\,\,{n_4}\,,{n_5})$$ is

Let $${n \ge 2}$$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is

Consider the set of eight vectors $$V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}$$. Three non-coplanar vectors can be chosen from v in $${2^p}$$ ways. Then p is

Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation: $$$\matrix{ {3x - y - z = 0} \cr { - 3x + z = 0} \cr { - 3x + 2y + z = 0} \cr } $$$

Then the number of such points for which $$x^2 + {y^2} + {z^2} \le 100$$ is