Final answer:
To find the coefficient of x²y⁶ in the expansion of (2x+y)⁸, we need to use the Binomial Theorem and the formula for the binomial coefficient. The coefficient is 28.
Step-by-step explanation:
To find the coefficient of x²y⁶ in the expansion of (2x+y)⁸, we need to use the Binomial Theorem. The general term in the expansion of (2x+y)⁸ is given by the formula:
C(8,k)(2x)^(8-k)(y)^k
where C(8,k) is the binomial coefficient. For the term with x²y⁶, we need to have k=6. Plugging in the values gives us:
C(8,6)(2x)^(8-6)(y)^6 = C(8,6)(2)²(x²)(y⁶)
Using the formula for the binomial coefficient, C(8,6) = 8! / (6!(8-6)!), we find that C(8,6) = 28. Therefore, the coefficient of x²y⁶ in the expansion of (2x+y)⁸ is 28.