Difference between revisions of "2016 UNCO Math Contest II Problems/Problem 8"
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== Problem == | == Problem == | ||
| + | Tree | ||
| + | |||
| + | |||
| + | Each circle in this tree diagram is to be assigned a value, chosen from a set <math>S</math>, in such a way that along every pathway down the tree, the assigned values never increase. That is, <math>A \ge B, A \ge C, C \ge D, C \ge E</math>, and <math>A, B, C, D, E \in S</math>. (It is permissible for a value in <math>S</math> to appear more than once.) | ||
| + | |||
| + | (a) How many ways can the tree be so numbered, using | ||
| + | only values chosen from the set <math>S = \{1, . . . , 6\}</math>? | ||
| + | |||
| + | (b) Generalize to the case in which <math>S = \{1, . . . , n\}</math>. 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 <math>n</math>. | ||
== Solution == | == Solution == | ||
Revision as of 01:38, 20 May 2017
Problem
Tree
Each circle in this tree diagram is to be assigned a value, chosen from a set 
, in such a way that along every pathway down the tree, the assigned values never increase. That is, 
, and 
. (It is permissible for a value in to appear more than once.)
(a) How many ways can the tree be so numbered, using
only values chosen from the set 
?
(b) Generalize to the case in which 
. 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 
.
Solution
See also
| 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 | ||
| All UNCO Math Contest Problems and Solutions | ||
[[Category:]]
