Question

explain how you can use the pattern for expanding a binomial to expand (x+y+z)^10

Answer 1

This is going to be long a messy in practice although simple in theory.

Consider first

(x + (y+z) )^10.

This is a binomial in x and (y+z). Expand this using standard binomial theory. Then you will need to expand all of the (y+z)^j terms.

Answer 2

To work a smaller example, consider (x+y+z)^3. Write this as
(x + (y+z) )^3.
By the binomial theorem this is equal to

x^3 + 3x^2(y+z) + 3x(y+z)^2 + (y+z)^3

Now expand the (y+z) terms

x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3

Answer 3

I would just use the multinomial expansion

Answer 4

That too. Either way, it's going to be tedious calculation.

Answer 5

although perhaps it would result in a slightly neater arrangement of terms.

x^3 + y^3 + z^3 + 3(x^2y + y^2z + z^2x + x^2z + y^2x + z^2y) + 6xyz