any technique or "trick" for finding the coefficient

davedave

Here is the polynomial expression

(x+x^2+x^3+x^4+x^5+x^6)^5

Each x term is raised to ascending powers of 1.

The entire sum in the brackets is raised to the 5th power.

Does anyone have any "special trick" for finding the coefficient of the term which contains x^19?

fourier jr

what about generating functions? there's another way to rewrite the finite geometric series inside the brackets, and that formula raised to the 5th power has another formula, and you can plug in the appropriate numbers to get the coefficient for x^19

Gib Z

Think about how would you write out the expansion normally? You would write out the 5 factors in 5 seperate brackets (..)(..)(..)(..)(..), and pick a term from each bracket to form a term on the RHS. How many ways can you pick those terms differently to get the exponents adding to 19? It's now a combinatorics problem.