MathDB

Let

k

be an integer,

k \geq 2

, and let

p_{1},\ p_{2},\ \ldots,\ p_{k}

be positive reals with p_{1} \plus{} p_{2} \plus{} \ldots \plus{} p_{k} \equal{} 1. Suppose we have a collection

\left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)

\left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)

\ldots

\left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)

k

-tuples of finite sets satisfying the following two properties: (i) for every

i

and every

j \neq j^{\prime}

, A_{i,j}\cap A_{i,j^{\prime}} \equal{} \emptyset, and (ii) for every

i\neq i^{\prime}

there exist

j\neq j^{\prime}

for which

A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset

. Prove that \sum_{b \equal{} 1}^{m}{\prod_{a \equal{} 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1.

2.1