Ownership Of Books

$n$ people in a town own $n$ different volumes of the encyclopedia in a specific manner. It is true that each resident owns $k$ different volumes; in addition, each volume is owned by $k$ different residents.

1. Show that for all $n$ and $k$ , that there exists a distribution of the volumes that satisfy the requirements.

For example, person 1 can own Volumes 1 and 2, person 2 can own volumes 2 and 5, person 3 can own volumes 4 and 5, etc.

2. Find the total number of books in the town.

3. Find the number of different ways the residents can own the books.

#Algebra #Generalize #GraphTheory #ProofProblems #Existence

Easy Math Editor

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Comments

Consider a complete bipartite graph L-R, each side containing n nodes. Connect each node in L with all n nodes in R by a link of unit capacity. Add two additional nodes S and T and connect S with all nodes in L by links each of capacity k and similarly, connect all nodes in R to the node T by links, each of capacity k. Send kn units of flow from S to T. Argue that min cut in this graph is kn and hence the flow is feasible. By integrality of capacities, each feasible flow gives a feasible solution.

Abhishek Sinha - 7 years ago

Although the proof method given below is very general (and somewhat advanced), it is not difficult to come up with an elementary existential proof for this particular problem. For this, consider the bipartite graph $L-R$ each side having $n$ nodes (denoting the set of persons and the set of books respectively). Number the nodes $\{1,2,\ldots,n\}$ on both sides. Now, connect a node numbered $i, 1\leq i \leq n$ from the $L$ set to $k$ nodes numbered $i+j\mod (n), j=0,1,2,\ldots k-1$ in the $R$ set. This assignment satisfies the given conditions.

Abhishek Sinha - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$