Final answer:
The coefficient of the y⁴q⁹ term in (2y + q)¹³ is found using the binomial theorem. The 10th term of the expansion gives the coefficient 13C9 multiplied by 2⁴, which equals 11440.
Step-by-step explanation:
To determine the coefficient of the y⁴q⁹ term in the expansion of (2y + q)¹³ using the binomial theorem, we will apply the formula for the nth term of a binomial expansion:
The general term is given by T(k+1) = nCk · a⁻¹⁴-k · b⁾k, where n is the total number of terms, a and b are the terms in the binomial, and k is the term number.
For the term with y⁴q⁹, k is equal to 9. So the term we're looking for is the 10th term (k+1).
Using the formula, we calculate:
T(10) = 13C9 · (2y)⁻⁹⁴-⁹ · q⁹ = 13C9 · 2⁴y⁴ · q⁹
13C9 is the number of ways to choose 9 elements from 13, which is also equal to 13C4 (since 13C9 = 13C13-9).
The binomial coefficient 13C9 is 13! / (9! · (13-9)!) = 13! / (9! · 4!) = 715.
So, multiplying this by 2⁴ (which is 16), we get the coefficient of the y⁴q⁹ term: 715 · 16 = 11440.