*I feel like the biggest idiot...*

**The answer is indeed 27.**

**We have to choose a path of exactly 8 edges i.e 9 vertices will be in the path.**

**First vertex must be 1, and Last vertex(i.e 9th vertex on the path) must be 4**

**Second vertex must be 2(as there is only one path from 1 i.e towards 2)**

**Third vertex could be anything(either 1 or 3 or 4 ie. total of 3 choices here)**

**Fourth vertex must be 2(there is only one path from 1 or 3 or 4 i.e toward 2)**

**Fifth vertex could be anything(i.e either 1 or 3 or 4 => total of 3 choices here)**

**Sixth vertex must be 2(same reason as for the fourth vertex)**

**7th vertex could be anything(1 or 3 or 4 so again 3 choices)**

**8th vertex must be 2**

**we've already defined 9th(the last vertex)**

**Now the path would look like:**

**(1)->(2)->(3 or 4 or 5)->(2)->(3 or 4 or 5)->(2)->(3 or 4 or 5)->(2)->(4)**

**so, simply we have 3*3*3 path = 27**

*I forgot to multiply by the last 3 and got the wrong answer;<*