Explanation Given: c k is the number of ways to make change for k dollars using $ 1 bills, $ 2 bills, $ 5 bills and $ 10 bills. Formula used: F1) ∑ k = 0 n x k = 1 − x n + 1 1 − x and for n → ∞ , ∑ k = 0 ∞ x k = 1 1 − x F2) ( 1 + x ) p = ∑ k = 0 + ∞ ( p k ) x k Calculation: Let the A 1 , A 2 , A 5 and A 10 be the total dollar bill that can be obtained using $ 1 bills, $ 2 bills, $ 5 bills and $ 10 bills respectively We know that, using $ 1 bills we can have any positive integer valued dollar bill It means corresponding generating function would be, 1 + x + x 2 + ⋅ ⋅ ⋅ Using $ 2 bills we can have any positive integer valued dollar bill in the multiples of 2 It means corresponding generating function would be, 1 + x 2 + x 4 + ⋅ ⋅ ⋅ Using $ 5 bills we can have any positive integer valued dollar bill in the multiples of 5 It means corresponding generating function would be, 1 + x 5 + x 10 ⋅ ⋅ ⋅ Using $ 10 bills we can have any positive integer valued dollar bill in the multiples of 10 It means corresponding generating function would be, 1 + x 10 + x 20 ⋅ ⋅ ⋅ Combining the generating function for each A i we get, ( 1 + x + x 2 + ⋅ ⋅ ⋅ ) ( 1 + x 2 + x 4 + ⋅ ⋅ ⋅ ) ( 1 + x 5 + x 10

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Chapter 8.4, Problem 19E

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The generating function for the sequence {ck} where ck is the number of ways to make change for k dollars using $1 bills, $2 bills, $5 bills and $10 bills

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Chapter 8.4, Problem 18E

Chapter 8.4, Problem 20E

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(a) (b) (c) (d) de unique? Answer the following questions related to the linear system x + y + z = 2 x-y+z=0 2x + y 2 3 rewrite the linear system into the matrix-vector form A = 5 Fuse elementary row operation to solve this linear system. Is the solution use elementary row operation to find the inverse of A and then solve the linear system. Verify the solution is the same as (b). give the null space of matrix A and find the dimension of null space. give the column space of matrix A and find the dimension of the column space of A (Hint: use Rank-Nullity Theorem).

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Chapter 8 Solutions

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

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The generating function for the sequence { c k } where c k is the number of ways to make change for k dollars using $ 1 bills, $ 2 bills, $ 5 bills and $ 10 bills

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To calculate: The generating function for the sequence {ck} where ck is the number of ways to make change for k dollars using $1 bills, $2 bills, $5 bills and $10 bills

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Chapter 8 Solutions

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The generating function for the sequence {ck} where ck is the number of ways to make change for k dollars using $1 bills, $2 bills, $5 bills and $10 bills