Adjacent arrangements I want to arrange n red and 5 orange balls in a way so all the orange balls are
adjacent to at least one other orange one.
I'm thinking that there either there will be a group of 5 orange balls, a group of 3 or a group of 2, so adjacency could take place. Where to from here? or should i take a different approach ?
 A: Your analysis is fine as far as it goes. Now consider the two cases. If the $5$ orange balls form a single group, there is one arrangement for every possible placement of that group in a line of $n$ red balls; how many places are there to put it?
If the orange balls are split $2$ and $3$ or $3$ and $2$, you have to pick $2$ of the positions in the line of $n$ red balls, and then you can insert the two groups of orange balls in $2$ ways, either $3$ and then $2$, or $2$ and then $3$, so you’ll get something of the form $2\binom{?}2$. Can you finish it from there?
A: I think the easiest way to think about it is this : The case where you have $5$ adjacent orange balls is actually just the case where you have $2$ orange-$3$ orange balls, but that those two groups are now sticked together. So replace the original problem by this : You have $n$ red balls, one $2$-group of orange balls and one $3$-group of orange balls, i.e. $3$ types of objects. Therefore there are $\begin{pmatrix} n + 2 \\ n,1,1 \end{pmatrix} = (n+2)(n+1)$ ways to arrange those. But when the orange balls are in a group of $5$, you're double counting, because it is the same thing if you place the group of $2$-orange balls to the left or to the right of the group of $3$-orange balls when they are next to each other, so in this count you remove once the number of cases where the $5$ orange balls are together, which gives you a final count of 
$$
\begin{pmatrix} n+2 \\ n,1,1 \end{pmatrix} - (n+1) = (n+2)(n+1) - (n+1) = (n+1)^2 = n^2 + 2n + 1.
$$
