Counting k-combinations excluding duplicate mathematically

Question

Let's say I have a multiset of integers a with a size n (here n = 10)

$$a = \{1, 1, 2, 2, 3, 4, 5, 6, 6, 10\}$$

I'd like to know the count of k-combinations I can obtain with a provided k value and excluding the duplicate combinations

From the dedicated article I know the formula to count those including duplicates is $\frac{n!}{k! (n - k)!}$, but I found no mention of a possible way to get a count while excluding them anywhere. Is there only a mathematical formula allowing to calculate it, or is the only solution to rely on some algorithm?

Answer 1

Possible strategies

As discussed in Number of k-combinations of a multiset, your problem of counting $k$-combinations $\mathcal C_k$ of some multiset $\{a_1^{(r_i)},a_2^{(r_i)},\dots,a_n^{(r_i)}\}$ where $(r_i)$ counts how many elements of kind $a_i$ are in there, is equivalent to counting the number of integer solutions to the equation

$$ x_1+x_2+x_3+\dots+x_n=k, 0\le x_i\le r_i, $$

where $x_i$ represents the number of times the element of $a_i$ kind was used in a combination.

Note that if $k=0,1$ then the answer is simply $1,n$. Otherwise, this problem does not have a closed form solution, but there are still different ways to solve it.

See answers given to extended stars-and-bars problem(where the upper limit of the variable is bounded). As demonstrated there, you can use for example the Inclusion–exclusion principle or look at coefficients of corresponding polynomial.

---

Applying possible strategies to your example

In the case of your example: $$\{1^{(2)},2^{(2)},3^{(1)},4^{(1)},5^{(1)},6^{(2)},10^{(1)}\}\implies n=7,$$ I will try out both mentioned answers to the previously linked question.

$$\text{I}.$$ We can for example use the Inclusion–exclusion principle: (see an answer from the linked post)

$$\begin{align} \mathcal C_k = \sum_{S\subseteq \{1,2,\dots,n\}}(-1)^{|S|}\binom{k+n-1-\sum_{i\in S}(r_i+1)}{n-1} \end{align}$$

Which is for corresponding $k$ expanded and simplified as:

$$\begin{align} \cal C_{2} = C_{8} &= \binom{8}{6} - 4\cdot\binom{6}{6} &= 24 \\ \cal C_{3} = C_{7} &= \binom{9}{6} - \left(3\cdot\binom{6}{6}+4\cdot\binom{7}{6}\right) &= 53 \\ \cal C_{4} = C_{6} &= \binom{10}{6} - \left(3\cdot\binom{7}{6}+4\cdot\binom{8}{6}\right) + 6\cdot\binom{6}{6} &= 83 \\ \cal C_{5} = C_{5} &= \binom{11}{6} - \left(3\cdot\binom{8}{6}+4\cdot\binom{9}{6}\right) + \left(12\cdot\binom{6}{6}+6\cdot\binom{7}{6}\right) &= 96 \\ \end{align}$$

Note that if total number of elements is $m$, then $\cal C_i = C_{m-i}$, which is $m=10$ in your example.

$$\text{II}.$$Alternatively, if you don't want to use Inclusion–exclusion principle, you can look at the coefficients of (see the other answer from the linked post):

$$ P(x) = \prod_{i=1}^n (1-x^{(r_i+1)}) = \sum_ic_ix^{e_i} $$

Then your answer becomes:

$$ {\cal{C}}_k = \sum_{i=0}^kc_i\binom{k-e_i+n-1}{n-1} $$

In the case of your example, the polynomial I got is:

$$P(x) = -x^{17}+\dots+18 x^{10}+11 x^9-11 x^8-18 x^7-x^6+12 x^5+6 x^4-3 x^3-4 x^2+1 $$

And when I substitute in the coefficients, I get the exact same equations as in previous method.

$$\text{III}.$$ Sometimes if your multiset is simple enough, you can use tricks or counting principles to get a result without needing to rely on things like previous two methods. Or if your multiset is of special kind, then there are nicer formulas. For example, if all $r_i$ are equal or if all $r_i$ are $\infty$.

If you want to explore such examples, you can search questions tagged [multisets] [combinations].

For example, the accepted answer to Unable to think about 10-combinations of a multiset is able to reduce the problem to a "Stars and bars" problem which does have a closed formula. The other answer there is also nice because it solves it graphically.

---

Remark

In case you wanted, you can easily print out every of those combinations for every $k$ with python.

from sympy.utilities.iterables import multiset_combinations
M = {1:2, 2:2, 3:1, 4:1, 5:1, 6:2, 10:1}
for k in range(sum(M.values())+1):

    print(k)
    for i,m in enumerate(multiset_combinations(M,k)):
        print(i+1,m)

The code gives the expected results for your example:

0
1 []
1
1 [1]
2 [2]
3 [3]
4 [4]
5 [5]
6 [6]
7 [10]
2
1 [1, 1]
2 [1, 2]
3 [1, 3]
4 [1, 4]
5 [1, 5]
6 [1, 6]
7 [1, 10]
8 [2, 2]
9 [2, 3]
10 [2, 4]
11 [2, 5]
12 [2, 6]
13 [2, 10]
14 [3, 4]
15 [3, 5]
16 [3, 6]
17 [3, 10]
18 [4, 5]
19 [4, 6]
20 [4, 10]
21 [5, 6]
22 [5, 10]
23 [6, 6]
24 [6, 10]
3
1 [1, 1, 2]
2 [1, 1, 3]
3 [1, 1, 4]
4 [1, 1, 5]
5 [1, 1, 6]
6 [1, 1, 10]
7 [1, 2, 2]
8 [1, 2, 3]
9 [1, 2, 4]
10 [1, 2, 5]
11 [1, 2, 6]
12 [1, 2, 10]
13 [1, 3, 4]
14 [1, 3, 5]
15 [1, 3, 6]
16 [1, 3, 10]
17 [1, 4, 5]
18 [1, 4, 6]
19 [1, 4, 10]
20 [1, 5, 6]
21 [1, 5, 10]
22 [1, 6, 6]
23 [1, 6, 10]
24 [2, 2, 3]
25 [2, 2, 4]
26 [2, 2, 5]
27 [2, 2, 6]
28 [2, 2, 10]
29 [2, 3, 4]
30 [2, 3, 5]
31 [2, 3, 6]
32 [2, 3, 10]
33 [2, 4, 5]
34 [2, 4, 6]
35 [2, 4, 10]
36 [2, 5, 6]
37 [2, 5, 10]
38 [2, 6, 6]
39 [2, 6, 10]
40 [3, 4, 5]
41 [3, 4, 6]
42 [3, 4, 10]
43 [3, 5, 6]
44 [3, 5, 10]
45 [3, 6, 6]
46 [3, 6, 10]
47 [4, 5, 6]
48 [4, 5, 10]
49 [4, 6, 6]
50 [4, 6, 10]
51 [5, 6, 6]
52 [5, 6, 10]
53 [6, 6, 10]
4
1 [1, 1, 2, 2]
2 [1, 1, 2, 3]
3 [1, 1, 2, 4]
4 [1, 1, 2, 5]
5 [1, 1, 2, 6]
6 [1, 1, 2, 10]
7 [1, 1, 3, 4]
8 [1, 1, 3, 5]
9 [1, 1, 3, 6]
10 [1, 1, 3, 10]
11 [1, 1, 4, 5]
12 [1, 1, 4, 6]
13 [1, 1, 4, 10]
14 [1, 1, 5, 6]
15 [1, 1, 5, 10]
16 [1, 1, 6, 6]
17 [1, 1, 6, 10]
18 [1, 2, 2, 3]
19 [1, 2, 2, 4]
20 [1, 2, 2, 5]
21 [1, 2, 2, 6]
22 [1, 2, 2, 10]
23 [1, 2, 3, 4]
24 [1, 2, 3, 5]
25 [1, 2, 3, 6]
26 [1, 2, 3, 10]
27 [1, 2, 4, 5]
28 [1, 2, 4, 6]
29 [1, 2, 4, 10]
30 [1, 2, 5, 6]
31 [1, 2, 5, 10]
32 [1, 2, 6, 6]
33 [1, 2, 6, 10]
34 [1, 3, 4, 5]
35 [1, 3, 4, 6]
36 [1, 3, 4, 10]
37 [1, 3, 5, 6]
38 [1, 3, 5, 10]
39 [1, 3, 6, 6]
40 [1, 3, 6, 10]
41 [1, 4, 5, 6]
42 [1, 4, 5, 10]
43 [1, 4, 6, 6]
44 [1, 4, 6, 10]
45 [1, 5, 6, 6]
46 [1, 5, 6, 10]
47 [1, 6, 6, 10]
48 [2, 2, 3, 4]
49 [2, 2, 3, 5]
50 [2, 2, 3, 6]
51 [2, 2, 3, 10]
52 [2, 2, 4, 5]
53 [2, 2, 4, 6]
54 [2, 2, 4, 10]
55 [2, 2, 5, 6]
56 [2, 2, 5, 10]
57 [2, 2, 6, 6]
58 [2, 2, 6, 10]
59 [2, 3, 4, 5]
60 [2, 3, 4, 6]
61 [2, 3, 4, 10]
62 [2, 3, 5, 6]
63 [2, 3, 5, 10]
64 [2, 3, 6, 6]
65 [2, 3, 6, 10]
66 [2, 4, 5, 6]
67 [2, 4, 5, 10]
68 [2, 4, 6, 6]
69 [2, 4, 6, 10]
70 [2, 5, 6, 6]
71 [2, 5, 6, 10]
72 [2, 6, 6, 10]
73 [3, 4, 5, 6]
74 [3, 4, 5, 10]
75 [3, 4, 6, 6]
76 [3, 4, 6, 10]
77 [3, 5, 6, 6]
78 [3, 5, 6, 10]
79 [3, 6, 6, 10]
80 [4, 5, 6, 6]
81 [4, 5, 6, 10]
82 [4, 6, 6, 10]
83 [5, 6, 6, 10]
5
1 [1, 1, 2, 2, 3]
2 [1, 1, 2, 2, 4]
3 [1, 1, 2, 2, 5]
4 [1, 1, 2, 2, 6]
5 [1, 1, 2, 2, 10]
6 [1, 1, 2, 3, 4]
7 [1, 1, 2, 3, 5]
8 [1, 1, 2, 3, 6]
9 [1, 1, 2, 3, 10]
10 [1, 1, 2, 4, 5]
11 [1, 1, 2, 4, 6]
12 [1, 1, 2, 4, 10]
13 [1, 1, 2, 5, 6]
14 [1, 1, 2, 5, 10]
15 [1, 1, 2, 6, 6]
16 [1, 1, 2, 6, 10]
17 [1, 1, 3, 4, 5]
18 [1, 1, 3, 4, 6]
19 [1, 1, 3, 4, 10]
20 [1, 1, 3, 5, 6]
21 [1, 1, 3, 5, 10]
22 [1, 1, 3, 6, 6]
23 [1, 1, 3, 6, 10]
24 [1, 1, 4, 5, 6]
25 [1, 1, 4, 5, 10]
26 [1, 1, 4, 6, 6]
27 [1, 1, 4, 6, 10]
28 [1, 1, 5, 6, 6]
29 [1, 1, 5, 6, 10]
30 [1, 1, 6, 6, 10]
31 [1, 2, 2, 3, 4]
32 [1, 2, 2, 3, 5]
33 [1, 2, 2, 3, 6]
34 [1, 2, 2, 3, 10]
35 [1, 2, 2, 4, 5]
36 [1, 2, 2, 4, 6]
37 [1, 2, 2, 4, 10]
38 [1, 2, 2, 5, 6]
39 [1, 2, 2, 5, 10]
40 [1, 2, 2, 6, 6]
41 [1, 2, 2, 6, 10]
42 [1, 2, 3, 4, 5]
43 [1, 2, 3, 4, 6]
44 [1, 2, 3, 4, 10]
45 [1, 2, 3, 5, 6]
46 [1, 2, 3, 5, 10]
47 [1, 2, 3, 6, 6]
48 [1, 2, 3, 6, 10]
49 [1, 2, 4, 5, 6]
50 [1, 2, 4, 5, 10]
51 [1, 2, 4, 6, 6]
52 [1, 2, 4, 6, 10]
53 [1, 2, 5, 6, 6]
54 [1, 2, 5, 6, 10]
55 [1, 2, 6, 6, 10]
56 [1, 3, 4, 5, 6]
57 [1, 3, 4, 5, 10]
58 [1, 3, 4, 6, 6]
59 [1, 3, 4, 6, 10]
60 [1, 3, 5, 6, 6]
61 [1, 3, 5, 6, 10]
62 [1, 3, 6, 6, 10]
63 [1, 4, 5, 6, 6]
64 [1, 4, 5, 6, 10]
65 [1, 4, 6, 6, 10]
66 [1, 5, 6, 6, 10]
67 [2, 2, 3, 4, 5]
68 [2, 2, 3, 4, 6]
69 [2, 2, 3, 4, 10]
70 [2, 2, 3, 5, 6]
71 [2, 2, 3, 5, 10]
72 [2, 2, 3, 6, 6]
73 [2, 2, 3, 6, 10]
74 [2, 2, 4, 5, 6]
75 [2, 2, 4, 5, 10]
76 [2, 2, 4, 6, 6]
77 [2, 2, 4, 6, 10]
78 [2, 2, 5, 6, 6]
79 [2, 2, 5, 6, 10]
80 [2, 2, 6, 6, 10]
81 [2, 3, 4, 5, 6]
82 [2, 3, 4, 5, 10]
83 [2, 3, 4, 6, 6]
84 [2, 3, 4, 6, 10]
85 [2, 3, 5, 6, 6]
86 [2, 3, 5, 6, 10]
87 [2, 3, 6, 6, 10]
88 [2, 4, 5, 6, 6]
89 [2, 4, 5, 6, 10]
90 [2, 4, 6, 6, 10]
91 [2, 5, 6, 6, 10]
92 [3, 4, 5, 6, 6]
93 [3, 4, 5, 6, 10]
94 [3, 4, 6, 6, 10]
95 [3, 5, 6, 6, 10]
96 [4, 5, 6, 6, 10]
6
1 [1, 1, 2, 2, 3, 4]
2 [1, 1, 2, 2, 3, 5]
3 [1, 1, 2, 2, 3, 6]
4 [1, 1, 2, 2, 3, 10]
5 [1, 1, 2, 2, 4, 5]
6 [1, 1, 2, 2, 4, 6]
7 [1, 1, 2, 2, 4, 10]
8 [1, 1, 2, 2, 5, 6]
9 [1, 1, 2, 2, 5, 10]
10 [1, 1, 2, 2, 6, 6]
11 [1, 1, 2, 2, 6, 10]
12 [1, 1, 2, 3, 4, 5]
13 [1, 1, 2, 3, 4, 6]
14 [1, 1, 2, 3, 4, 10]
15 [1, 1, 2, 3, 5, 6]
16 [1, 1, 2, 3, 5, 10]
17 [1, 1, 2, 3, 6, 6]
18 [1, 1, 2, 3, 6, 10]
19 [1, 1, 2, 4, 5, 6]
20 [1, 1, 2, 4, 5, 10]
21 [1, 1, 2, 4, 6, 6]
22 [1, 1, 2, 4, 6, 10]
23 [1, 1, 2, 5, 6, 6]
24 [1, 1, 2, 5, 6, 10]
25 [1, 1, 2, 6, 6, 10]
26 [1, 1, 3, 4, 5, 6]
27 [1, 1, 3, 4, 5, 10]
28 [1, 1, 3, 4, 6, 6]
29 [1, 1, 3, 4, 6, 10]
30 [1, 1, 3, 5, 6, 6]
31 [1, 1, 3, 5, 6, 10]
32 [1, 1, 3, 6, 6, 10]
33 [1, 1, 4, 5, 6, 6]
34 [1, 1, 4, 5, 6, 10]
35 [1, 1, 4, 6, 6, 10]
36 [1, 1, 5, 6, 6, 10]
37 [1, 2, 2, 3, 4, 5]
38 [1, 2, 2, 3, 4, 6]
39 [1, 2, 2, 3, 4, 10]
40 [1, 2, 2, 3, 5, 6]
41 [1, 2, 2, 3, 5, 10]
42 [1, 2, 2, 3, 6, 6]
43 [1, 2, 2, 3, 6, 10]
44 [1, 2, 2, 4, 5, 6]
45 [1, 2, 2, 4, 5, 10]
46 [1, 2, 2, 4, 6, 6]
47 [1, 2, 2, 4, 6, 10]
48 [1, 2, 2, 5, 6, 6]
49 [1, 2, 2, 5, 6, 10]
50 [1, 2, 2, 6, 6, 10]
51 [1, 2, 3, 4, 5, 6]
52 [1, 2, 3, 4, 5, 10]
53 [1, 2, 3, 4, 6, 6]
54 [1, 2, 3, 4, 6, 10]
55 [1, 2, 3, 5, 6, 6]
56 [1, 2, 3, 5, 6, 10]
57 [1, 2, 3, 6, 6, 10]
58 [1, 2, 4, 5, 6, 6]
59 [1, 2, 4, 5, 6, 10]
60 [1, 2, 4, 6, 6, 10]
61 [1, 2, 5, 6, 6, 10]
62 [1, 3, 4, 5, 6, 6]
63 [1, 3, 4, 5, 6, 10]
64 [1, 3, 4, 6, 6, 10]
65 [1, 3, 5, 6, 6, 10]
66 [1, 4, 5, 6, 6, 10]
67 [2, 2, 3, 4, 5, 6]
68 [2, 2, 3, 4, 5, 10]
69 [2, 2, 3, 4, 6, 6]
70 [2, 2, 3, 4, 6, 10]
71 [2, 2, 3, 5, 6, 6]
72 [2, 2, 3, 5, 6, 10]
73 [2, 2, 3, 6, 6, 10]
74 [2, 2, 4, 5, 6, 6]
75 [2, 2, 4, 5, 6, 10]
76 [2, 2, 4, 6, 6, 10]
77 [2, 2, 5, 6, 6, 10]
78 [2, 3, 4, 5, 6, 6]
79 [2, 3, 4, 5, 6, 10]
80 [2, 3, 4, 6, 6, 10]
81 [2, 3, 5, 6, 6, 10]
82 [2, 4, 5, 6, 6, 10]
83 [3, 4, 5, 6, 6, 10]
7
1 [1, 1, 2, 2, 3, 4, 5]
2 [1, 1, 2, 2, 3, 4, 6]
3 [1, 1, 2, 2, 3, 4, 10]
4 [1, 1, 2, 2, 3, 5, 6]
5 [1, 1, 2, 2, 3, 5, 10]
6 [1, 1, 2, 2, 3, 6, 6]
7 [1, 1, 2, 2, 3, 6, 10]
8 [1, 1, 2, 2, 4, 5, 6]
9 [1, 1, 2, 2, 4, 5, 10]
10 [1, 1, 2, 2, 4, 6, 6]
11 [1, 1, 2, 2, 4, 6, 10]
12 [1, 1, 2, 2, 5, 6, 6]
13 [1, 1, 2, 2, 5, 6, 10]
14 [1, 1, 2, 2, 6, 6, 10]
15 [1, 1, 2, 3, 4, 5, 6]
16 [1, 1, 2, 3, 4, 5, 10]
17 [1, 1, 2, 3, 4, 6, 6]
18 [1, 1, 2, 3, 4, 6, 10]
19 [1, 1, 2, 3, 5, 6, 6]
20 [1, 1, 2, 3, 5, 6, 10]
21 [1, 1, 2, 3, 6, 6, 10]
22 [1, 1, 2, 4, 5, 6, 6]
23 [1, 1, 2, 4, 5, 6, 10]
24 [1, 1, 2, 4, 6, 6, 10]
25 [1, 1, 2, 5, 6, 6, 10]
26 [1, 1, 3, 4, 5, 6, 6]
27 [1, 1, 3, 4, 5, 6, 10]
28 [1, 1, 3, 4, 6, 6, 10]
29 [1, 1, 3, 5, 6, 6, 10]
30 [1, 1, 4, 5, 6, 6, 10]
31 [1, 2, 2, 3, 4, 5, 6]
32 [1, 2, 2, 3, 4, 5, 10]
33 [1, 2, 2, 3, 4, 6, 6]
34 [1, 2, 2, 3, 4, 6, 10]
35 [1, 2, 2, 3, 5, 6, 6]
36 [1, 2, 2, 3, 5, 6, 10]
37 [1, 2, 2, 3, 6, 6, 10]
38 [1, 2, 2, 4, 5, 6, 6]
39 [1, 2, 2, 4, 5, 6, 10]
40 [1, 2, 2, 4, 6, 6, 10]
41 [1, 2, 2, 5, 6, 6, 10]
42 [1, 2, 3, 4, 5, 6, 6]
43 [1, 2, 3, 4, 5, 6, 10]
44 [1, 2, 3, 4, 6, 6, 10]
45 [1, 2, 3, 5, 6, 6, 10]
46 [1, 2, 4, 5, 6, 6, 10]
47 [1, 3, 4, 5, 6, 6, 10]
48 [2, 2, 3, 4, 5, 6, 6]
49 [2, 2, 3, 4, 5, 6, 10]
50 [2, 2, 3, 4, 6, 6, 10]
51 [2, 2, 3, 5, 6, 6, 10]
52 [2, 2, 4, 5, 6, 6, 10]
53 [2, 3, 4, 5, 6, 6, 10]
8
1 [1, 1, 2, 2, 3, 4, 5, 6]
2 [1, 1, 2, 2, 3, 4, 5, 10]
3 [1, 1, 2, 2, 3, 4, 6, 6]
4 [1, 1, 2, 2, 3, 4, 6, 10]
5 [1, 1, 2, 2, 3, 5, 6, 6]
6 [1, 1, 2, 2, 3, 5, 6, 10]
7 [1, 1, 2, 2, 3, 6, 6, 10]
8 [1, 1, 2, 2, 4, 5, 6, 6]
9 [1, 1, 2, 2, 4, 5, 6, 10]
10 [1, 1, 2, 2, 4, 6, 6, 10]
11 [1, 1, 2, 2, 5, 6, 6, 10]
12 [1, 1, 2, 3, 4, 5, 6, 6]
13 [1, 1, 2, 3, 4, 5, 6, 10]
14 [1, 1, 2, 3, 4, 6, 6, 10]
15 [1, 1, 2, 3, 5, 6, 6, 10]
16 [1, 1, 2, 4, 5, 6, 6, 10]
17 [1, 1, 3, 4, 5, 6, 6, 10]
18 [1, 2, 2, 3, 4, 5, 6, 6]
19 [1, 2, 2, 3, 4, 5, 6, 10]
20 [1, 2, 2, 3, 4, 6, 6, 10]
21 [1, 2, 2, 3, 5, 6, 6, 10]
22 [1, 2, 2, 4, 5, 6, 6, 10]
23 [1, 2, 3, 4, 5, 6, 6, 10]
24 [2, 2, 3, 4, 5, 6, 6, 10]
9
1 [1, 1, 2, 2, 3, 4, 5, 6, 6]
2 [1, 1, 2, 2, 3, 4, 5, 6, 10]
3 [1, 1, 2, 2, 3, 4, 6, 6, 10]
4 [1, 1, 2, 2, 3, 5, 6, 6, 10]
5 [1, 1, 2, 2, 4, 5, 6, 6, 10]
6 [1, 1, 2, 3, 4, 5, 6, 6, 10]
7 [1, 2, 2, 3, 4, 5, 6, 6, 10]
10
1 [1, 1, 2, 2, 3, 4, 5, 6, 6, 10]