Answer :
Final answer:
The coefficient of x^9y^3 in the expansion of (x + y)^12 is 220. For (x + 2y)^12 and (2x - 3y)^12, the coefficients calculated do not match any of the given choices.
Explanation:
The problem is asking us to find the coefficient of x^9y^3 in the expansion of three binomials - (x + y)^12, (x + 2y)^12, and (2x - 3y)^12. To solve this problem, we need to use the Binomial Theorem - a result of expanding the power of a binomial.
In the case of (x + y)^12, the coefficient is given by the combination 12 choose 9 because the exponent of x is 9 in that term. Therefore, the combination 12C9 is equal to 220, which is choice A for (a).
For the case of (x + 2y)^12, the coefficient is given by 12C9 multiplied by the coefficient of y, which is 2, raised to the power of 3. Therefore, the coefficient is 12C9 * 2^3 = 220 * 8 = 1760, which is not one of the choices given.
For the case of (2x - 3y)^12, the coefficient is given by 12C9 multiplied by the coefficients of x and y, which are 2 and -3 respectively, raised to the powers of 9 and 3. Therefore, the coefficient is 12C9 * 2^9 * (-3)^3 = 220 * 512 * -27 = -30481920, which is not one of the choices given.
Learn more about Binomial Theorem here:
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