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Motivated by the many roles that hook lengths play in mathematics, we studythe distribution of the number of $t$-hooks in the partitions of $n$. We provethat the limiting distribution is normal with mean $\mu_t(n)\sim\frac{\sqrt{6n}}{\pi}-\frac{t}{2}$ and variance $\sigma_t^2(n)\sim\frac{(\pi^2-6)\sqrt{6n}}{2\pi^3}.$ Furthermore, we prove that the distributionof the number of hook lengths that are multiples of a fixed $t\geq 4$ inpartitions of $n$ converge to a shifted Gamma distribution with parameter$k=(t-1)/2$ and scale $\theta=\sqrt{2/(t-1)}.$

Distributions of Hook lengths in integer partitions