For integers n and r, let $$\left( {\matrix{ n \cr r \cr } } \right) = \left\{ {\matrix{ {{}^n{C_r},} & {if\,n \ge r \ge 0} \cr {0,} & {otherwise} \cr } } \right.$$ The maximum value of k for which the sum $$\sum\limits_{i = 0}^k {\left( {\matrix{ {10} \cr i \cr } } \right)\left( {\matrix{ {15} \cr {k - i} \cr } } \right)} + \sum\limits_{i = 0}^{k + 1} {\left( {\matrix{ {12} \cr i \cr } } \right)\left( {\matrix{ {13} \cr {k + 1 - i} \cr } } \right)} $$ exists, is equal to _________.

The coefficient of x256 in the expansion of (1 $-$ x)101 (x2 + x + 1)100 is :

For the natural numbers m, n, if ${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$ and ${a_1} = {a_2} =…

If b is very small as compared to the value of a, so that the cube and other higher powers of ${b \over a}$ can be neglected in the…

The lowest integer which is greater than ${\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}}$ is ______________.

If the co-efficient of $x^9$ in ${\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}}$ and the co-efficient of $x^{-9}$ in ${\left(…