Combinations

jabrown

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Oct 22, 2021
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The question is:
How many combinations of numbers can you make with the following numbers: 2, 2, 5, 5, 3

My only way of solving it was to write out every combination:
1 number combinations (select 1 number from the group w/o replacement): 2, 3, or 5
2 number combinations (select 2 numbers from the group w/o replacement): 2-2, 2-3, 2-5, 5-5, or 5-3
3 number combinations (select 3 numbers from the group w/o replacement): 2-2-5, 2-2-3, 2-5-5, 2-5-3, or 5-5-3
4 number combinations (select 4 numbers from the group w/o replacement): 2-2-5-5, 2-2-5-3, or 2-5-5-3
5 number combinations (select 5 numbers from the group w/o replacement): 2-2-5-5-3

That adds up to a total of 17 unique combinations. There's got to be a faster way, with less room for error, to do that. Any ideas?
 
The question is:
How many combinations of numbers can you make with the following numbers: 2, 2, 5, 5, 3

My only way of solving it was to write out every combination:
1 number combinations (select 1 number from the group w/o replacement): 2, 3, or 5
2 number combinations (select 2 numbers from the group w/o replacement): 2-2, 2-3, 2-5, 5-5, or 5-3
3 number combinations (select 3 numbers from the group w/o replacement): 2-2-5, 2-2-3, 2-5-5, 2-5-3, or 5-5-3
4 number combinations (select 4 numbers from the group w/o replacement): 2-2-5-5, 2-2-5-3, or 2-5-5-3
5 number combinations (select 5 numbers from the group w/o replacement): 2-2-5-5-3

That adds up to a total of 17 unique combinations. There's got to be a faster way, with less room for error, to do that. Any ideas?
Suppose that we have the five symbols 2,2,5,5,32, 2, 5, 5, 3. We are to make strings of length five using each of those symbols and only once.
That can be done in 5!(2!)2=30\dfrac{5!}{(2!)^2}=30 ways.
However, that is what the question asks but is not what your listing of examples show.
The number of ways to rearrange the letters MISSISSIPPIMISSISSIPPI is 11!(4!)2(2!)=34650\dfrac{11!}{(4!)^2(2!)}=34650 ways.

 
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