The determinantal formula gives an expression of Specht modules by permutational modules in the Grothendieck ring of modules over the symmetric group S_n. It was shown by Zelevinskii in "Resolutions, dual pairs and character formulas" (J. Funktsional. Anal. i Prilozhen. Vol. 21 p. 74-75) that for each Specht module over a field of zero characteristic there is a coresolution by permutational modules that realises this expression. We construct permutational coresolutions for Specht modules over a field of positive characteristic.