Final answer:
To find the coefficient of x²y¹⁵ in the expansion of (5x²y)⁾ using the binomial theorem, we calculate the 16th term because the powers of x and y add up to 17. The formula (⁾C¹⁵ × 5²) gives us the coefficient after simplifying, which results in 816 × 25.
Step-by-step explanation:
To find the coefficient of x²y¹⁵ in the expansion of (5x²y)⁾ using the binomial theorem, we first identify 'a' as 5x² and 'b' as y. In the expansion of (a + b)⁾, the general series expansions term for kth term is given by the formula: Tᵗʳ = ⁾Cᵗ × a⁾⁻ᵗ × bᵗ where ⁾Cᵗ represents the combination of n items taken k at a time.
To find the coefficient of x²y¹⁵, we need the term where the power of x is 2 and the power of y is 15. This is the 16th term in the expansion, because we start counting from 0. Using the above formula, we calculate:
- ⁾C¹⁵ = 18! / (15! × (18-15)!)
- a³ = (5x²)³
- b¹⁵ = y¹⁵
The coefficient is just the number in front of x²y¹⁵, which is ⁾C¹⁵ × (5²) and can be simplified further. We calculate ⁾C¹⁵ as (18 × 17 × 16) / (3 × 2 × 1) = 816. Then, we multiply by (5²), which is 25. Therefore, the coefficient of x²y¹⁵ is 816 × 25.