We compute three-body depletion interactions in a hard-sphere mixture withi
n the framework of density-functional theory and by considering the infinit
e dilution limit of the functional. The results look very accurate and show
three-body interactions much smaller than the pair depletion ones, reveali
ng that these are strongly influenced by correlations and have a decay leng
th similar to the two-body depletion potential. The results are compared wi
th the predictions of the Asakura-Oosawa model for the triplet interactions
.