Homework 6,7,8,9 - Homework Solutions 5.1.8 a 4 3 2 = 14 b...

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Homework Solutions 5.1.8 a) 4 · 3 + 2 = 14 b) 14 2 = 196 c) 196 - 2 2 = 192 d) 4 · 3(2 + 3 · 2) + 2(1 + 4 · 3) = 122 5.1.9 a) ( 4 1 )( 47 1 ) = 188 b) ( 12 1 )( 47 1 ) + ( 1 1 )( 48 1 ) = 612 5.1.31 1 100 100000 + 10 1000 * 100000 + 100 10000 100000 + 1000 100000 100000 = 4000 5.2.16 a) ( 48 1 ) ( 52 5 ) = 0 . 00001847 b) ( 13 1 )( 48 1 ) ( 52 5 ) = 0 . 0002401 c) ( 13 2 )( 4 2 ) 2 ( 11 1 )( 4 1 ) ( 52 5 ) = 0 . 04754 d) ( 2 1 )( 13 2 )( 4 3 )( 4 2 ) ( 52 5 ) = 0 . 001441 e) ( 10 1 )( 4 1 ) 5 ( 52 5 ) = 0 . 00394 f) ( 13 5 )( 4 1 ) 5 ( 52 5 ) = 0 . 5071 1
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5.2.30 4! ( 7 3 ) = 840 5.2.37 4 · 3! 7! 2!2! = 30240 5.3.13 ( 2 1 ) 6! 2!2! = 360 5.3.15 ( 10 6 ) [ ( 6 2 ) 8! 2!2! + ( 6 1 ) 8! 3! ] = 40219200 5.4.2 a) 4 1 6 = 4294967296 b) ( 4 2 )( 2 2 )( 16 6 )( 10 6 )( 4 2 )( 2 2 ) = 60540480 c) 16! 4!4!4!4! = 63063000 5.4.14 99 n =4 ( n - 1 3 ) = ( 99 4 ) = 3764376 6.1.2 a) (1 + x + x 2 + x 3 + x 4 ) 5 b) ( x + x 2 + x 3 + x 4 ) 3 c) ( x 2 + x 4 + x 6 )( x 3 + x 5 ))( x 2 + x 3 + x 4 + x 5 + x 6 ) 2 d) (1 + x + x 2 + . . . ) 4 e) ( x + x 2 + . . . )( x + x 3 + x 5 + . . . )( x + x 3 ) 2
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6.1.4 a) (1 + x + x 2 + x 3 + x 4 + x 5 ) 5 b) ( x 3 + x 4 + x 5 + x 6 ) 4 c) ( x + x 2 + x 3 + . . . ) 7 d) (1 + x + x 2 + x 3 + x 4 + x 5 )(1 + x + x 2 + . . . ) 2 6.1.19 (1 + x + x 2 + . . . ) 2 ( x + x 2 + x 3 + . . . ) 4 . We want the coefficient of x 1 5, and in general x n - 5 . 6.2.4 g ( x ) = ( x + x 2 + x 3 + x 4 + x 5 )( x 2 + x 3 + x 4 + . . . ) 5 = x 1 1(1 + x + x 2 + x 3 + x 4 )(1 + x + x 2 . . . ) 5 = x 1 1(1 - x 5 ) 1 (1 - x ) 6 . The coefficient of x 18 is then ( 12 7 ) - ( 7 2 ) = 771. 6.2.22 g ( x ) = ( x + x 2 + x 3 + x 4 + x 5 + x 6 ) 10 = x 10 (1 + x + x 2 + x 3 + x 4 + x 5 ) = x 10 (1 - x 6 ) 10 1 (1 - x ) 10 .
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