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1. Use generating functions to prove that, for n 1, the numb...

1. Use generating functions to prove that, for n 1, the number of integer parti- tions of n in which only odd parts may be repeated is equal to the number of integer partitions of n in which no part appears more than 3 times. 2. We know from class that when t = 1 the upper bound (n-1) given by th...

1. Show that the number of partitions of k into n parts of s...

1. Show that the number of partitions of k into n parts of size at most m equals the number of partitions of mn - k into no more than n parts of size at most m - 1. 2. How does the number of partitions of k relate to the number of partitions of k + 1 whose smallest part is one?

1. What is the generating function for the number of ways to...

1. What is the generating function for the number of ways to pass out k pieces of candy from an unlimited supply of identical candy to n children (where n is fixed) so that each child gets between three and six pieces of candy (inclusive)? Use the fact that (1+x+x2+x3(1 -x)=1-x4 - = to find a f...

1. Use the inductive definition of an to prove that (ab)&...

1. Use the inductive definition of an to prove that (ab)" = an bb for all nonnegative integers n. 3. A hydrocarbon molecule is a molecule whose only atoms are either carbon atoms or hydrogen atoms. In a simple molecular model of a hydrocarbon, a carbon atom will bond to exactly four oth...

(1) (a) Simplify the following expression without usi...

(1) (a) Simplify the following expression without using a computer or doing it the long way. Explain your answer. (x+18-8(x+17+28(x+1)6-56(x+15+70(x+14-56(x+1)³ - - - + 28(x + 1) - 8(x + 1) +1 (b) Write find a formula for x7 as a linear combination of powers of (x - 2). That ...

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