(4) The left-hand side is the number of ways of choosing r balls from n+1. Suppose one ball is coloured blue (and the others aren't). Now explain why the right-hand side is the number of ways of picking r balls including the blue one plus the number of ways of picking r balls excluding the blue one.

(5) OK, this isn't true. But the inductive step works. So what's gone wrong? It's not true for n=1. This is why it's absolutely vital that you check the starting point!

(6) The inductive hypothesis is that 4^k + 6k - 1 is divisible by 9. That is, 4^k+6k-1=9m for some integer m. Now use this to get an expression for 4^k that you can substitute into 4^k+1+6(k+1)-1. Alternatively, what is 4(4^k+6k-1)-18k+9? This is more elegant, but perhaps harder to spot without practice!