5 Burgers and Fries

Suppose there are $n$ dishes. Consider a $n \times 5$ table where the rows represent the dishes and the columns represent the people. In cell $(i,j)$ , place a $1$ if person $j$ ordered dish $i$ and place a zero otherwise. We will double count $\mathcal{M}$ , the number of pairs of $1$ in the same row. Since each row has $3$ ones, $\mathcal{M} = n \binom{3}{2}$ . Since every pair of people ordered $k$ common dishes, $\mathcal{M} = k \binom{5}{2}$ . So $n \binom{3}{2} = k \binom{5}{2} \implies 3n = 10k.$ Since $n,k$ are both positive integers, the minimal value would be $\boxed{n = 10}, k = 3$ . A construction is as shown: $\begin{pmatrix} 1 & 1 & 1 & 0 & 0\\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \end{pmatrix}$

Consider 5 people are $P=(p_1,p_2,p_3,p_4,p_5)$

Consider there are $n$ dishes such as $D=(d_1,d_2,d_3,.........,d_n)$ .

So,there is a mapping between $f: P \rightarrow D$ where $f(p_i)=d_j$ for some $i,j$ .Now we will consider the lines by which $p_i's,d_j's$ are connected.According to the question each dish $d_i$ is exactly ordered by $3$ people.So from each dish there will be $3$ edges which will connected to $P$ side.So,total no of lines/edges $3n$ .

Now each pair $p_i,p_j$ has $k$ dishes in common.Let $p_i$ ordered $d_1$ i.e. $f(p_i)=d_1$ and also $p_j$ ordered $d_1$ i.e. $f(p_j)=d_1$ .Each common dish will shares $2$ edges.So for $k$ common dishes there will be $2k$ edges.Total number of pairs

${10 \choose 2}=10$ .Total edges $10 \times 2 k=20 k$ .

Consider a pair $p_i,p_j$ both of them order a common dish $d_1$ .So an edge is counted from $p_i$ to $d_1$ and another an edge $p_j$ to $d_1$ .Again consider another pair $p_i,p_k$ both of them orders a common dish $d_1$ (Total three persons ordered dish $d_1$ ).So an edge is counted from $p_i$ to $d_1$ and another edge is counted from $p_k$ to $d_1$ ..So, in counting the edge $p_i$ to $d_1$ is counted twice.This will be counted for every edges.So, $3n$ edges are counted extra.

$20k-3n=3n \\ \implies 6n=20k \\ \implies 3n=10k$ .

If minimal $k=1$ then minimal $n=10$