Homework_1_Solution - STAT/MATH 418 Solution to Homework 1...

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Unformatted text preview: STAT/MATH 418 Solution to Homework 1 Chapter 1 6. 7 . 7 7 7 = 2401 7. (a) 6! = 720 9. (b) 2 = 72 3! 3! (c) 4!3! = 144 (d) 6 1 = 72 3 2 2 1 (12)! = 27, 720 6!4! 10 10 12 12 15. 199,584 ( or the answer could be if we consider "paired them off".) = 5! 5 5 5 5 21. 7! = 35 . Each path is a linear arrangement of 4r's and 3u's (r for right and u for up). For instance 3!4! 4 n1 n2 n3 ( x1 ) ( 2 x2 ) ( 3x3 ) n n1 + n2 + n3 = 4 1 , n2 , n3 the arrangement r, r, u, u, r, r, u specifies the path whose first 2 steps are to the right, and final step is up. 26. ( x1 + 2 x2 + 3 x3 ) = 4 4 4 4 0 4 0 0 0 4 0 0 4 = ( ( ( x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) 4,0,0 0, 0,0, 4,0 4 4 3 4 3 4 0 1 0 0 1 3 1 + ( ( ( x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) 3,1,0 3,0,1 0,3,1 4 1 4 1 4 x 3 0 0 3 0 1 3 + ( ( ( x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3 x3 ) 1,3,0 1,0,3 0,1,3 4 2 4 2 4 0 2 0 0 2 2 2 + ( ( ( x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3 x3 ) + x1 ) ( 2 x2 ) ( 3 x3 ) 2, 2,0, 0, 2,0 2 2, 2 4 1 4 1 4 1 1 2 2 1 1 2 + x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) + x1 ) ( 2 x2 ) ( 3x3 ) ( ( ( 1,1, 2 1, 2,1 1,1, 2 27. 12 12! = represents the number of possible divisions of 12 distinct objects into 3 3, 4, 5 3!4!5! distinct groups of respective sizes 3, 4 and 5. (use the multinomial coefficients) ...
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This note was uploaded on 06/28/2008 for the course MATH 418 taught by Professor Arnold,stevenfdong,yuexiao during the Summer '08 term at Pennsylvania State University, University Park.

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