Discrete Mathematics with Graph Theory (3rd Edition) 206

# Discrete Mathematics with Graph Theory (3rd Edition) 206 -...

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204 Solutions to Exercises 2. (a + 4b)5 a 5 + 5a 4 (4b) + lOa 3 (4b)2 + 10a 2 (4b)3 + 5a(4b)4 + (4b)5 = a 5 + 20a 4 b + 160a 3 b 2 + 640a 2 b 3 + 1280ab 4 + 1024b 5 3. (a) (2x3 - y2)8 = (2x 3 )8 + m (2x 3 )1 (_y2) + @(2x 3 )6( _y2)2 +(~)(2x3)5(_y2)3 + (~)(2x3)4(_y2)4 + (~)(2x3)3(_y2)5 + (:)(2x 3 )2(_y2)6 + (~)(2x3)(_y2)1 + (_y2)8 ( 3 1 )8 (c) 2x - x 2 256x 24 - 8(128x2I )y2 + 28(64x I8 )y4 - 56(32x I5 )y6 + 70(16x I2 )y8 - 56(8x 9 )yI0 + 28( 4x 6 )y12 _ 8(2x 3 )yI4 + yI6 256x 24 - 1024x2Iy2 + 1792x I8 y4 - 1792x I5 y6 + 1120x 12 y8 _ 448x 9 yI0 + 112x 6 y12 _ 16x3yI4 + yI6 (2x 3 )8 + (~) (2x 3 )7 (-x 2 ) + (~) (2x 3 )6( _X 2 )2 +(~)(2X3)5(_X2)3 + (~)(2x3)4(_x2)4 + (~)(2x3)3(_X2)5 +(:) (2x 3 )2( _X 2 )6 + (~) (2x 3 )( -x 2 )1 + (_x 2 )8 256x 24 - 8(128x 23 ) + 28(64x 22 ) - 56(32x2I) +70(16x20) - 56(8xI9) + 28(4x I8 ) - 8(2xI7) + x I6 256x 24 - 1024x 23 + 1792x 22 - 1792x 2I + 1120x 2o -448x 19 + 112x 18 _ 16x I7 + x I6 (2x3)8 + (8) (2x3)1 (_~) + (8) (2x3)6( _~)2 + (8) (2x3)5( _~)3 1 x2 2
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Unformatted text preview: x2 3 x2 +(8)(2x3)4(_~)4 + (8)(2x3)3(_~)5 + (8)(2x3)2(_~)6 4 x2 5 x2 6 x2 +(8)(2x3)(-~f + (_~)8 7 x2 x2 256x 24 -8(128x2I)~ + 28(64xI8)~ -56(32xI5)~ x 2 x4 x 6 +70(16x I2 )-is--56(8x 9 )xk + 28(4x 6 )xh -8(2x3) ~4 + ~6 X X 256x 24 _ 1024xI9 + 1792xI4 _ 1792x 9 + 1120x 4 _ 448 + 112 _ ~ + _I_ X x 6 xlI x I6 4. [BB] C3 2 ) (X 3 )9(_2y2)3 = -1760x 27 y6 5. (a) x 20 , 20x 19 , (2~)xI8y2 = 190x I8 y2 (c) [BB] The binomial coefficients appear in the order e~), eI0),. .. . The coefficient appearing in the seventh term is e60) , so the seventh term is (2~) x I4 y6 = 38, 760X14y6. Similarly, the fifteenth term is (~~)x6yI4 = 38,760X 6 yI4. (d) (~) = 77,520 6. (a) [BB] 17 terms (b) [BB] There is a middle term (since 16 is even), namely, C8 6 ) (2x)8( _y)8 = 12,870(256)x 8 y8 = 3,294,720x 8 y8....
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