Cascade triggers whenever you cast a spell with cascade from your hand or any zone — the trigger condition is simply that you cast the spell. Paying an alternative cost such as flashback does not prevent the cascade ability from triggering, because cascade cares only about the act of casting, not how you paid for it (CR 702.85a).
The key distinction is between casting a spell and paying for it. Alternative costs like flashback replace what you pay, but the spell is still being cast, so any triggered abilities that trigger on casting — including cascade — will still see the event and trigger normally (CR 118.9).
There is one important nuance: cascade itself exiles cards and lets you cast one of them without paying its mana cost. If you cascade into a spell that also has flashback, you could pay flashback instead of casting it for free, but that is a separate choice unrelated to the original trigger.
Example: You cast Captured Sunlight (cascade) from your graveyard using its flashback cost. Cascade still triggers. You exile cards until you reveal a nonland card with mana value 3 or less, then may cast it without paying its mana cost.
Citations: CR 702.85a defines cascade's trigger condition as casting the spell; CR 118.9 confirms alternative costs replace payment but do not change the fact that a spell is being cast.
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