- #1

scoutfai

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Given [tex]n[/tex] objects in a set. All the objects can be categorized into [tex]k[/tex] groups such that no two objects from different groups are identical. Objects in the same group are indistinguishable from each other within the group. Number of objects in each group is as shown in the following:

1st group has [tex]m_{1}[/tex] objects;

2nd group has [tex]m_{2}[/tex] objects;

3rd group has [tex]m_{3}[/tex] objects;

... ...

[tex]k^{th}[/tex] group has [tex]m_{k}[/tex] objects.

What is the number of possible combination if one was to pick [tex]r[/tex] objects from the set?

What is the number of possible permutation if one was to pick [tex]r[/tex] objects from the set?

Example:

AAA BBBB CC DDD EE FF G H I JJ

There are 21 ([tex]n=21[/tex]) alphabets above. These alphabets can be categorized into 10 ([tex]k=10[/tex]) groups. The number of alphabets in each group is as shown:

1st group (A), [tex]m_{1}=3[/tex]

2nd group (B), [tex]m_{2}=4[/tex]

3rd group (C), [tex]m_{3}=2[/tex]

4th group (D), [tex]m_{4}=3[/tex]

5th group (E), [tex]m_{5}=2[/tex]

6th group (F), [tex]m_{6}=2[/tex]

7th group (G), [tex]m_{7}=1[/tex]

8th group (H), [tex]m_{8}=1[/tex]

9th group (I), [tex]m_{9}=1[/tex]

10th group (J), [tex]m_{10}=2[/tex]

What is the number of possible combination if one was to pick 5 ([tex]r=5[/tex]) objects from this set of alphabets?

What is the number of possible permutation if one was to pick 5 ([tex]r=5[/tex]) objects from this set of alphabets?

For this kind of question, if every parameters were given its value to me, then I can solve for the answer (a series of lengthy calculation, one has to consider all possibilities one by one). But when the parameters remain as unknown, I am unable to derive a general solution (a formula).

I suspect that this is a kind of problem that can be represented analytically, but one can not solve it analytically. Just like the equation [tex]e^{x}sin x=0[/tex] , the problem can be represented analytically, but one can not express x analytically and can only be solved numerically.

Any thought?