when we number a carbon chain and encounter a double bond and a triple bond , the priority is given to the double bond .
While giving the number to the double bond , please make sure that you give the double bond the least possible number .
As we know that there are two carbons attached to the double bond . So we give the double bond the number which is the least (that is the number given to lesser carbon atom )
What i am saying is , you cant give the numbers 2 and 4 to double bonds until you have the possibility of 1 and 3 therefore the name cannot end with 2,4 diene .
Second thing , numbering the double bond is to be checked after numbering the parent chain . And the numbering of parent chain is all correct.
So the name given in the solution is correct .