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The objects can be expressed as: Y Y G G R R BE BE BK BK P P O O O.

I want to be able to select duplicates and any number of objects from $1$ to $15$ so $1$ combination would be O,O. Another would be Y,Y,G,G,R,R,BE,BE,BK,BK,P,P,O,O,O and so on.

user2868467
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  • duplicates is unclear: do you mean the letters are indistinguishable or that you can select more than one of a letter? – qwr Mar 02 '14 at 07:03

3 Answers3

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Since you have $15$ objects and you can select any number of them with repetition, then you have $$15^1+15^2+15^3+...+15^{15}$$ different combinations possible. In other words, a very large number.

homegrown
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Since duplicates are allowed, every element is in one of two states: in your selection or not in your selection. Subtracting the case where you select nothing, this is $2^{15} - 1$.

qwr
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Answer:

If I understand your problem correctly, you would want to treat each of these as distinguishable objects ( when you see with duplicates) and any combination of 2 objects to 15 objects from your example.

O,O will contain two distinguishable objects O1,O2 and O2,O1 are indeed the same O,O but are with duplicates.

Selecting 1 object from a set of 15 objects = $15$

Selecting a string of two objects from a set of 15 objects = $15^2$

Similarly, selecting a string of three objects from a set of 15 objects = $15^3$

and goes to selecting a string of 15 objects from a set of 15 objects = $15^{15}$

Again a word of caution, all these strings will have duplicates under the line of reasoning that I laid out initially and it seems like it is OK with you.

Thus the total number of combinations

= $$15+15^2+15^3+15^4+\cdots+15^{15} = \frac{(15^{16}-1)}{(15-1)}$$ $$=6,568,408,355,712,890,000/14$$ $$=469,172,025,408,064,000 $$

Satish Ramanathan
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