Answer: 17 cards
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Step-by-step explanation:
The best case scenario is that you pull 5 cards from the same suit in a row (eg: 5 spades in a row). However, we have to plan for the worst case scenario that way we can guarantee to have 5 cards in one suit without having to look at the cards.
The worst case scenario is that you pull cards such that you get one suit at a time (eg: spade, heart, club, diamond). On the 5th draw, we finally get a card that repeats a suit.
If we repeat the sequence {spade, heart, club, diamond} in that order a total of two times, then we'll have 8 cards with 2 cards per suit.
Doing that sequence four times we'll have 4*4 = 16 cards and 4 cards per suit. That 17th card will repeat a suit a fifth time and this is when we'll be guaranteed to have at least 5 cards of one suit.