Effects like Cutting Words (PHB, p. 54) and Spirit Shield (XGtE, p. 10) reduce damage by a roll, irrespective of damage type (PHB 196). However, single damage rolls can have multiple damage types, as confirmed (for wording) by Jeremy Crawford in this tweet.
How do you know which part of the damage is reduced?
This could actually make a difference to the amount of damage taken if, for example, the creature taking damage is resistant to one or more of the damage types involved.
Cutting Words: ... When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature's roll.
Barny, an Ice Devil (MM, p. 75) hits Bobby, rolling 9 slashing damage and 11 cold damage.
Bobby's ally, Vera, is a level 10 bard of the college of Lore, and uses her Cutting Words ability on the attack's damage roll. She rolls a 9, which is subtracted from the Barny's damage roll.
Bobby is wearing his Boots of the Winterlands (DMG, p. 156), giving him resistance to cold.
We know that "resistance and then vulnerability are applied after all other modifiers to damage" (PHB, p. 197) and both slashing damage and cold damage are treated as "one big damage roll" (see the JC tweet), so is all the 9 slashing damage removed by cutting words, and then the cold damage halved for 5 total damage, or is 9 of the cold damage removed by cutting words, leaving 2 to be halved for a total of 1 + 9 = 10 damage taken by Bobby?
Who makes this decision?
When an effect (like cutting words) reduces damage without specifying type from a source containing multiple damage types, what part of the damage is reduced?