late 14c., permutacioun, "interchange, concurrent change; exchange of one thing, position, condition, etc., for another," from Old French permutacion "change, shift" (14c.), from Latin permutationem (nominative permutatio) "a change, alteration, revolution," noun of action from past participle stem of permutare "change thoroughly, exchange," from per "thoroughly" (see per) + mutare "to change" (from PIE root *mei- (1) "to change, go, move"). The sense of "a linear arrangement of objects resulting from a change of their order" is by 1710, originally in mathematics.

Permutation differs from combination in this, that in the latter there is no reference to the order in which the quantities are combined, whereas in the former this order is considered, and consequently the number of permutations always exceeds the number of combinations. [Century Dictionary]

late 14c., from Old French transmutacion "transformation, change, metamorphosis" (12c.), from Late Latin transmutationem (nominative transmutatio) "a change, shift," noun of action from past-participle stem of Latin transmutare "change from one condition to another," from trans "across, beyond; thoroughly" (see trans-) + mutare "to change" (from PIE root *mei- (1) "to change, go, move"). A word from alchemy.

mid-14c., "change the form of" (transitive), from Old French transformer (14c.), from Latin transformare "change in shape, metamorphose," from trans "across, beyond" (see trans-) + formare "to form" (see form (v.)). Intransitive sense "undergo a change of form" is from 1590s. Related: Transformed; transforming.

1797, "to cause (a state, etc.) to undergo a (political) revolution, effect a change in the political constitution of;" see revolution + -ize. Transferred sense of "change a thing completely and fundamentally, effect radical change in" is by 1799. Related: Revolutionized; revolutionizing.

late 14c., alteracioun, "change, transformation, action of altering," from Old French alteracion "change, alteration" (14c.), and directly from Medieval Latin alterationem (nominative alteratio), noun of action from past-participle stem of Late Latin alterare "to change," from Latin alter "the other (of the two)," from PIE root *al- (1) "beyond" + comparative suffix -ter (as in other). Meaning "change in character or appearance" is from 1530s; that of "change in ready-made clothes to suit a customer's specifications" is from 1901. Related: Alterations.

1898 in the modern sense of "change the order of" (earlier "to change, alter, 16c. but obsolete thereafter), from Latin permutatus, past participle of permutare "change thoroughly, exchange" (see permutation). "Probably regarded by those who use it as a back-formation from permutation" [OED]. Compare permute. Related: Permutated; permutating.

late 14c., "to change (something), make different in some way," from Old French alterer "to change, alter," from Medieval Latin alterare "to change," from Latin alter "the other (of the two)," from PIE root *al- (1) "beyond" + comparative suffix -ter (as in other). Intransitive sense "to become otherwise" first recorded 1580s. Related: Altered; altering.