Assn6Soln - 1 The proposed solution must satisfy the...

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1. The proposed solution must satisfy the recurrence, i.e. kc n - 9 kc n - 1 + 14 kc n - 2 = c n kc 2 - 9 kc + 14 k = c 2 k ( c 2 - 9 c + 14) = c 2 We can choose k = c 2 c 2 - 9 c +14 (and hence the solution of the required form exists), unless c 2 - 9 c + 14 = 0, i.e., unless c = 2 or c = 7. 2. 8.1.8: This is the same as the number of solutions to y 1 + y 2 + y 3 + y 4 = 39 with 0 y i 15 for i = 1 , 2 , 3 , 4. Consider the set of all ( 42 3 ) solutions to this equation for that the variables are nonnegative (but possibly greater than 15), and let p i be the property that y i 16. Then N ( p k ) (for k = 1 , 2 , 3 , 4) is the same as the number of solutions to z 1 + z 2 + z 3 + z 4 = 39 with y k 16, which is ( 26 3 ) . Similarly, N ( p k p j ) = ( 10 3 ) for any k and j such that 1 k < j 4. By principle of inclusion and exclusion, the number of solutions to the original equation is ( 42 3 ) - 4 ( 26 3 ) + 6 ( 10 3 ) . 8.1.13: Let p 1 be the property that the permutation contains the substring spin , p 2 the property that it contains game , p 3 that it contains path and p 4 that it contains net . To determine N ( p 1
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