Question

According to the binomial theorem, what would be the coefficient of

x⁴y²in the expansion of (xy)⁶?
a) 15
b) 20
c) 30
d) 60

Answer 1

The coefficient of x⁴y² in the expansion of (xy)⁶ is 15, following the binomial theorem. It corresponds to the term 15a⁴b² using the a and b from the original expression (a = x, b = y). Thus, option (a) is the correct answer.

Step-by-step explanation:

According to the binomial theorem, the coefficient of x⁴y² in the expansion of (xy)⁶ can be determined using the formula for the binomial expansion which states that (a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶.

Each term in this expansion corresponds to the number of ways to choose a certain number of items from a larger set, which is denoted as n choose k or C(n, k). Specifically, the coefficient we need is the coefficient of the term with a⁴b². Looking at the formula, this is represented by 15a⁴b² (where a = x and b = y in the original question).

Therefore, the coefficient of x⁴y² is 15, which corresponds with option (a).