Consider the expansion of (𝑥 + 2𝑦 + 3𝑧 + 4𝑤)18?a) Find the coefficient of 𝑥6𝑦3𝑧4𝑤5.b) Find the sum of all coefficients in the expansion.c) How many terms are in the expansion?

a) The coefficient of a term in the expansion of (x + 2y + 3z + 4w)^18 can be found using the multinomial theorem. The coefficient of x^6y^3z^4w^5 is given by:

(18 choose 6,3,4,5) * (x^6 * (2y)^3 * (3z)^4 * (4w)^5)

= (18! / (6!3!4!5!)) * (x^6 * 2^3 * y^3 * 3^4 * z^4 * 4^5 * w^5)

= 18564 * x^6 * y^3 * z^4 * w^5

So, the coefficient of x^6y^3z^4w^5 is 18564.

b) The sum of all coefficients in the expansion of (x + 2y + 3z + 4w)^18 is the same as the value of the expansion when x = y = z = w = 1. So, the sum of all coefficients is (1 + 2 + 3 + 4)^18 = 10^18.

c) The number of terms in the expansion of (x + 2y + 3z + 4w)^18 is given by the formula (n + r - 1) choose r, where n is the number of variables (4 in this case) and r is the power (18 in this case). So, the number of terms is (4 + 18 - 1) choose 18 = 21 choose 18 = 1330.