Correct spelling


Correct spelling, explanation: programs is the correct form in American English, while programmes is the correct one in British English, however, it’s never programms. Coming from Latin programma, British programmes was simplified in American English, just as many other words.

Definition of programs:
noun, plural of program, an outline of the order to be followed, often on some event; also: a printed leaflet with the outline during some event
Programs of the conference are available at the entrance and on the table at the back of the room.
I really liked the festival, the programs every day were grasping.

Incorrect spelling


Incorrect spelling, explanation: it looks like programms is a mistake made by combining American and British versions of the spelling (programs and progrmmes). The correct form is programs (or programmes in the UK), but never programms. The misspelling is probably made due to the confusion with two different possible spellings.