# Challenge problems in combinatorics theory

1) For this problem you have flags, which are distinct, and flagpoles, which are also distinct. The flags are placed on the flagpole in an order, so that if a red flag is on top of a blue flag, that's different than a blue flag being on top of a red flag. Assume the flagpoles are tall enough to hold all the flags.

a. List all the ways to place 3 flags on 2 poles, so that each pole has at least one flag. Invent your own notation for the problem.

b. How many ways are there to place 10 flags on 6 flagpoles if flagpoles are allowed to be empty?

c. How many ways are there to place 10 flags on 6 flagpoles if flagpoles are not allowed to be empty?

d. How many ways are there to place n flags on k flagpoles if flagpoles are allowed to be empty? Assume n and k are natural numbers.

e. How many ways are there to place 10 flags on 6 flagpoles if flagpoles are not allowed to be empty? Assume n and k are natural numbers.

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#### Solution Summary

Challenging problems in combinatorics theory, based on counting flags and flagpoles arranged in a specified order.