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hw2-1-sol

# hw2-1-sol - CS 202 Data Structures and Discrete Mathematics...

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Unformatted text preview: CS 202: Data Structures and Discrete Mathematics II Fall 2010 Homework 2-1 (Due September 22 in Class) No Late Submissions 1. (14=3+3+3+5 points) Binomial Theorem: (a) Find the 6th term in the expansion of ( x- 1 x 3 ) 10 . Then find the coefficient of 1 x 14 in this expansion. Solution: By Binomial theorem, x- 1 x 3 10 = 10 X k =0 C (10 ,k ) x k (- 1 /x 3 ) 10- k = 10 X k =0 C (10 ,k )(- 1) 10- k x k x 3(10- k ) = 10 X k =0 C (10 ,k )(- 1) 10- k 1 x 30- 3 k- k The 6th term in the expansion is the term for k = 5 which is C (10 , 5)(- 1) 10- 5 1 x 30- 15- 5 =- 252 x 10 From the above expansion, 1 /x 14 will show up in the term where 30- 3 k- k = 14, which means k = 4. Thus, the coefficient of 1 /x 14 is C (10 , 4)(- 1) 10- 4 = C (10 , 4) = 210 . (b) Use the binomial theorem twice to expand ( x + y + z ) 3 . Solution: ( x + y + z ) 3 = ( x + ( y + z )) 3 = 3 ( y + z ) x 3 + 3 1 ( y + z ) x 2 + 3 2 ( y + z ) 2 x + 3 3 ( y + z ) 3 x = x 3 + 3( y + z ) x 2 + 3( y + z ) 2 x + ( y + z ) 3 = x 3 + 3( y + z ) x 2 + 3 x 2 z y 2 + 2 1 z 1 y 1 + 2 2 z 2 y + 3 z y 3 + 3 1 z 1 y 2 + 3 2 z 2 y 1 + 3 3 z 3 y = x 3 + 3( y + z ) x 2 + 3 x ( y 2 + 2 zy + z 2 ) + ( y 3 + 3 zy 2 + 3 z 2 y + z 3 ) = x 3 + 3 x 2 y + 3 x 2 z + 3 xy 2 + 6 xyz + 3 xz 2 + y 3 + 3 zy 2 + 3 z 2 y + z 3 = x 3 + y 3 + z 3 + 3 x 2 z + 3 xy 2 + 3 xz 2 + 3 zy 2 + 3 z 2 y + 6 xyz 2-1-1 CS 202 Homework 2-1 Due September 22 in Class (c) Use the binomial theorem to prove that C ( n, 0) + C ( n, 1)2 + C ( n, 2)2 2 + + C ( n,n )2 n = 3 n Solution: Notice that 2 + 1 = 3 and 1 k = 1 for any k . Therefore, we can rewrite the equation above as C ( n, 0)1 n + C ( n, 1)2 1 n- 1 + C ( n, 2)2 2 1 n- 2 + + C ( n,n )2 n 1 = (1 + 2) n = 3 n . (d) Prove the identity in (c) using a combinatorial counting argument. Solution: We will show that both sides count the number of strings of length n where there are three choices for each letter (say, a , b , and c ). Clearly, the right hand side counts the number of such strings since each position has three independent choices and there are n positions, so by the rule of product we get 3 n ....
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hw2-1-sol - CS 202 Data Structures and Discrete Mathematics...

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