AP 08 - Feb 13

# AP 08 - Feb 13 - 1 Math 2602 M1/M2 Spring 2007 Additional...

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1 Math 2602 M1/M2 Spring 2007 Additional Problems 8 Binomial expansion and Inclusion-Exclusion Principle 1 Tuesday, Feb 13th 1. Find the coefficient of x 2 y 8 z 9 w 6 in the expansion of ( x - 2 y 2 + 5 z 3 - 2 w 2 ) 12 . 2. (a) Find the formula for n k =1 k 2 n k x n - k y k by differentiating ( x + y ) n with respect to suitable variable and multiplying the resulting expression by this suitable variable. (b) Compute 2007 k =1 k 2 2007 k 4 k 5 2007 by choosing x, y and n suitably in part (a). 3. In how many ways can the 26 letters of the alphabet be permuted so that none of the patterns pen, fur, or coal occurs ? 4. Find the number of integer solution to x 1 + x 2 + x 3 = 15 with the restriction 0 x 1 4 , 0 x 2 6 and 0 x 3 8 . 5. How many social security numbers (9-digit) have each of the digits 2 and 5 appearing at least once ? 6. Eight married couples are to be seated at a circular table. In how many ways can they arrange themselves so that no wife sits next to her husband ? (here, two seated arrangements are considered the same if one is a rotation of the other.)

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