I cannot make sense of the expansion hint in ii) and how that could relate to solving this problem, and I would like an explanation of how these 2 hints are useful, preferably further hint before a solution.
(I solved the first part of the question through combinations & permutations and got an answer of 369)
The answer to part (ii) is the same as the answer to "find the coefficient of $x^6$ in the expansion of $$(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-\cdots)^3$$" where inside the bracket is the infinite series $$1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-\cdots =\sum_{n=0}^\infty(-1)^n(n+1)x^n=\frac1{(1+x)^2}.$$