Multiset of Strings

You are given three integers $$$n$$$, $$$k$$$ and $$$f$$$.

Consider all binary strings (i. e. all strings consisting of characters $$$0$$$ and/or $$$1$$$) of length from $$$1$$$ to $$$n$$$. For every such string $$$s$$$, you need to choose an integer $$$c_s$$$ from $$$0$$$ to $$$k$$$.

A multiset of binary strings of length exactly $$$n$$$ is considered beautiful if for every binary string $$$s$$$ with length from $$$1$$$ to $$$n$$$, the number of strings in the multiset such that $$$s$$$ is their prefix is not exceeding $$$c_s$$$.

For example, let $$$n = 2$$$, $$$c_{0} = 3$$$, $$$c_{00} = 1$$$, $$$c_{01} = 2$$$, $$$c_{1} = 1$$$, $$$c_{10} = 2$$$, and $$$c_{11} = 3$$$. The multiset of strings $$$\{11, 01, 00, 01\}$$$ is beautiful, since:

• for the string $$$0$$$, there are $$$3$$$ strings in the multiset such that $$$0$$$ is their prefix, and $$$3 \le c_0$$$;
• for the string $$$00$$$, there is one string in the multiset such that $$$00$$$ is its prefix, and $$$1 \le c_{00}$$$;
• for the string $$$01$$$, there are $$$2$$$ strings in the multiset such that $$$01$$$ is their prefix, and $$$2 \le c_{01}$$$;
• for the string $$$1$$$, there is one string in the multiset such that $$$1$$$ is its prefix, and $$$1 \le c_1$$$;
• for the string $$$10$$$, there are $$$0$$$ strings in the multiset such that $$$10$$$ is their prefix, and $$$0 \le c_{10}$$$;
• for the string $$$11$$$, there is one string in the multiset such that $$$11$$$ is its prefix, and $$$1 \le c_{11}$$$.

Now, for the problem itself. You have to calculate the number of ways to choose the integer $$$c_s$$$ for every binary string $$$s$$$ of length from $$$1$$$ to $$$n$$$ in such a way that the maximum possible size of a beautiful multiset is exactly $$$f$$$.