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probability randomly selecting a pair of integers

Source: Polish MO Recond Round 1983 p6

September 9, 2024
probabilitycombinatoricsnumber theory

Problem Statement

For a given number n n , let us denote by pn p_n the probability that when randomly selecting a pair of integers k,m k, m satisfying the conditions 0≤k≤m≤2n 0 \leq k \leq m \leq 2^n (the selection of each pair is equally probable) the number (mk)\binom{m}{k} will be even. Calculate lim⁡n→∞pn \lim_{n\to \infty} p_n .
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