2016 UNCO Math Contest II Problems/Problem 8

Problem

Tree

Each circle in this tree diagram is to be assigned a value, chosen from a set $S$ , in such a way that along every pathway down the tree, the assigned values never increase. That is, $A \ge B, A \ge C, C \ge D, C \ge E$ , and $A, B, C, D, E \in S$ . (It is permissible for a value in $S$ to appear more than once.)

(a) How many ways can the tree be so numbered, using only values chosen from the set $S = \{1, . . . , 6\}$ ?

(b) Generalize to the case in which $S = \{1, . . . , n\}$ . Find a formula for the number of ways the tree can be numbered.

For maximal credit, express your answer in closed form as an explicit algebraic expression in $n$ .

2016 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2016 UNCO Math Contest II Problems/Problem 8

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