Piling camels (piling in bins 1/4)
In the board game Camel Up , there are 5 different camels to play with and there are 16 squares on the board on which the camels can stay. Whenever there are more than one camel on the same square, they are put one on top of another, forming a pile that can be 5 camels tall! The last camel to arrive is put on top and this order is very important.
How many ways are there to dispose the 5 camels on the 16 squares?
Part of the Piling distinct objects in bins set.
The answer is 1860480.
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1 solution
If the camels were all identical, piling them or not wouldn't change the number of ways to do it, and that would be ( ( 5 1 6 ) ) = ( 5 2 0 ) = 1 5 ′ 5 0 4 . Now, each camel has a distinct position, so if we want to paint them with 5 colors, there is 5 ! = 1 2 0 ways to do it.
Answer is then ( ( 5 1 6 ) ) ⋅ 5 ! = 1 5 ′ 5 0 4 ⋅ 1 2 0 = 1 ′ 8 6 0 ′ 4 8 0 .
( ( k n ) ) is the number of ways to choose one of n elements k times, without order, with possibility to choose the same element several times. That is the number of ways to place k indistinguishable balls into n labeled urns.
( ( k n ) ) = ( k n + k − 1 ) See stars and bars .