Question

Find the coefficient of x2y32215 in the expansion of ⁽ˣ ⁺ ²ʸ ⁻ ³ᶻ ⁺ ²ᵗ ⁺ ⁵⁾¹⁶ The result should include binomial coefficient. Do NOT compute the final result just provide it as factorial, binomial etc.

Answer 1

The coefficient of x^2y^32215 in the expansion of (x + 2y - 3z + 2t + 5)^16 is zero because the exponent of y exceeds the total exponent of the expansion, which is 16. Therefore, such a term does not exist in the expansion due to the principles of the binomial theorem and its extension, the multinomial theorem.

Step-by-step explanation:

To find the coefficient of x2y32215 in the expansion of (x + 2y - 3z + 2t + 5)16, we use binomial theorem. The general term in the expansion of (a + b)n is given by:

Tk+1 = C(n, k) × an-k × bk

where C(n, k) is the binomial coefficient n choose k, which can also be represented as n!/(k!(n-k)!). As we have more than two terms in the expression, we need to apply the multinomial theorem, which is an extension of the binomial theorem.

Since we seek the coefficient of x2y32215, we must identify the term in the expansion where x is raised to the power of 2 and y is raised to the power of 32215. However, since the power of y exceeds the total power of the expansion (16), the term x2y32215 does not exist in the expansion, and the coefficient is thus zero.