Combinations

[jabrown]

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?

 

[pka]

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 [imath]2, 2, 5, 5, 3[/imath]. We are to make strings of length five using each of those symbols and only once.
That can be done in [imath]\dfrac{5!}{(2!)^2}=30[/imath] 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 [imath]MISSISSIPPI[/imath] is [imath]\dfrac{11!}{(4!)^2(2!)}=34650[/imath] ways.

[imath][/imath][imath][/imath]