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> **How many trees on $\{1,2,3,4,5,6,7\}$ have a vertex of degree 2 ?**
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**Attempt -**

It feels like an inclusion exclusion problem (using kailey's code) , let's define $|A\_i| \Rightarrow $ vertex $i$ is of degree $1$.

$|Ai\, \cup A\_j$| - vertices $i,j$ are of degree $1$ and so on.

So following my calculation we might get - 

$7 \cdot 5\cdot 6^4 - \binom{7}{2}\binom{5}{2}\cdot2\cdot5^3+ \binom{7}{3}\binom{5}{3}\cdot3!\cdot4^2 - \binom{7}{4}\binom{5}{4}\cdot4!\cdot3 + \binom{7}{5}5!$ 

**What do you think? thank you !**