Combinatorics Problem: How Many Ways Can I Put K Birds into M Cages?

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To determine the number of ways to place K distinguishable birds into M distinguishable cages, each bird can independently choose one of the M cages. This results in M choices for each of the K birds, leading to a total of M^K combinations. The problem allows for empty cages, meaning some cages may not contain any birds. The discussion emphasizes that the labeling of both birds and cages simplifies the calculation. The final answer is M raised to the power of K, reflecting the straightforward nature of the problem.
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Homework Statement


I have K birds to put into M cages. How many ways can I do this(no limit on the amount of birds in a cage - there can be empty cages)?


Unfortunately I haven't had a combinatorics course (and my friends who have apparently have forgotten all of it...) so I'm a little lost.

A push in the right direction would be most appreciated.
 
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Are the cages numbered/labeled?
Are the birds numbered/labeled?
 
Both the birds and the cages are labeled
 
hi alex! :smile:

if the birds are all distinguishable, you just line them up, put a letter from 1 to M round their necks, and read off the overall number …

how many different overall numbers are there? :wink:​
 
well, in that case the answer is really trivial and straightforward. Give it another try.
 

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