Permanganate ion is reduced to #MnO_2#, and ammonia is oxidized to nitrate ion. Can you write a balanced chemical equation to represent the redox process?

You really have to make a meal out of these redox equations.

Permanganate ion (STRONGLY COLOURED PURPLE) could be reduced to almost colourless #Mn^(2+)#....

#stackrel(+VII)[MnO_4]^(-) +8H^(+) + 5e^(-) rarr Mn^(2+) + 4H_2O# #(i)#

But here it is reduced to #MnO_2#.

#stackrel(+VII)[MnO_4]^(-) +4H^(+) + 3e^(-) rarr stackrel(+IV)"MnO"_2 + 2H_2O# #(ii)#

Ammonia is OXIDIZED to nitrate anion.......(there is no associated colour change; ammonia pen and inks very badly, whereas ammonium ion/salt is odourless, but I do not suggest you use this method to differentiate them!)

#stackrel(-III)"NH"_3 +3H_2O rarr (stackrel(+V)"NO"_3)^(-) +9H^(+)+ 8e^(-)# #(iii)#

Now if I have done my sums right; #(i)#, #(ii)#, and #(iii)# ARE STOICHIOMETRICALLY BALANCED WITH RESPECT TO MASS AND CHARGE. If they are not, then they cannot be accepted as a representation of chemical reality. I think they are balanced, and so the final redox equation eliminates the electrons, and we take the sum...........#8xx(i)+5xx(iii)#:

#8MnO_4^(-) +5NH_3 + cancel(15H_2O) + cancel(64)19H^(+) rarr 8Mn^(2+) + 5NO_3^(-) +cancel(45H^(+))+ cancel(32)17H_2O#

And we cancel out common reagents to give (finally!):

#8MnO_4^(-) +5NH_3 +19H^(+) rarr 8Mn^(2+) + 5NO_3^(-) + 17H_2O#

Even despite the whack coefficients this is stoichiometrically balanced AS IS ABSOLUTELY REQUIRED...........

But you specified the reduction of permanganate to #MnO_2#, so this is ...........#3xx(i)+5xx(iii)#:

#8MnO_4^(-) +3NH_3 +5H^(+)rarr 8MnO_2(s) + 3NO_3^(-) +7H_2O#

But I note (well eventually I did!) that you specified BASIC conditions; all I have to do is add #5xxHO^-# to each side of the equation............and eliminate the waters..........

#8MnO_4^(-) +3NH_3 rarr 8MnO_2(s) + 3NO_3^(-) +5HO^(-) + 2H_2O#

Manganese metal has a particularly rich redox chemistry. For another example of this redox manifold, see here. See here for redox reactions in general.