2 MnVIIO4- + 5 NIIIO2- + 6 H3O+ 5 NVO3- + 2 (MnII)2+ + 9 H2O

This is an oxidation-reduction (redox) reaction:

2 MnVII + 10 e- 2 MnII (reduction)

5 NIII - 10 e- 5 NV (oxidation)

Change the oxidation states