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In the expansion of ∑_(r=0)⁵⁰ (50)/(r)(2x-3)^r(2-x)^(50-r), what is the coefficient of x²⁵?

1. (50)/(25)
2. -(50)/(30)
3. (50)/(30)
4. -(50)/(25)

Correct answer: -(50)/(25)

Solution

By the binomial theorem the sum equals ((2x-3)+(2-x))^50 = (x-1)^50. The coefficient of x^25 is C(50,25)*(-1)^25 = -C(50,25).

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