The total number of two digit numbers 'n', such that 3ⁿ + 7ⁿ is a multiple of 10, is __________.

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 x⁴ in the expansion of
(1 + x + x² + x³)⁶ in powers of x, is ______.

The natural number m, for which the coefficient of x in the binomial expansion of

$${\left( {{x^m} + {1 \over {{x^2}}}} \right)^{22}}$$ is 1540, is .............