222m1 - MATH 222[A01 Fall 2011 First Midterm Test...

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MATH 222 [A01] Fall 2011 First Midterm Test Instructor: Dr. Jing Huang October 12, 2011 3:30 - 4:20pm Instructions: This question paper has six pages plus cover. For each question, write out your solution/answer carefully in the space provided. Use the back of pages if necessary. Marks will be deducted for incomplete or poorly presented solutions/answers. The Sharp EL-510R calculator is the only calculator permitted. No other aids (such as books, notes, formula lists, or scratch paper) are allowed. Questions Value Marks 1 4 2 6 3 5 4 6 5 5 6 4 Total 30 Name: Id: .../1
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1 1. (a) State the binomial theorem and the Pascal’s identity involving the binomial coefficients. [2] (b) Determine the coefficient of x 2 y 5 in the expansion of ( x + 2 y + 1) 10 . [2] .../2
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2 2. How many strings of length 14 are there, using symbols from { U,V,I,C } , such that (a) each string contains exactly two U ’s, three V ’s, four I ’s, and five C ’s? [2] (b) each string contains exactly two U ’s which are not consecutive, three V ’s, four I ’s, and five C ’s? [2] (c) each string either begins with U or ends with C (and no other restrictions)? [2] .../3
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3 3. How many integer solutions are there to the equation
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