If $$\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}} $$, then k is equal to _____________.

Let the sixth term in the binomial expansion of $${\left( {\sqrt {{2^{{{\log }_2}\left( {10 - {3^x}} \right)}}} + \root 5 \of {{2^{(x - 2){{\log }_2}3}}} } \right)^m}$$ in the increasing powers of $$2^{(x-2) \log _{2} 3}$$, be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is __________.

The sum of the common terms of the following three arithmetic progressions.

$$3,7,11,15, \ldots ., 399$$,

$$2,5,8,11, \ldots ., 359$$ and

$$2,7,12,17, \ldots ., 197$$,

is equal to _____________.

Number of integral solutions to the equation $$x+y+z=21$$, where $$x \ge 1,y\ge3,z\ge4$$, is equal to ____________.