Determine the coefficient, a, for the term ax5y7 of the binomial expansion of (2x+y)12.
Question
Determine the coefficient, a, for the term ax<sup>5</sup>y<sup>7</sup> of the binomial expansion of (2x+y)<sup>12</sup>.
Solution
The binomial expansion of (2x+y)^12 can be represented by the binomial theorem as follows:
(2x+y)^12 = Σ (from k=0 to 12) [12Ck * (2x)^(12-k) * (y)^k]
The term ax^5y^7 corresponds to the case where k=7 (because the power of y is 7). So, we substitute k=7 into the binomial theorem:
a = 12C7 * (2x)^(12-7) * (y)^7 a = 792 * (2x)^5 * y^7 a = 792 * 32x^5 * y^7 a = 25344x^5 * y^7
So, the coefficient a for the term ax^5y^7 of the binomial expansion of (2x+y)^12 is 25344.
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