The minimum number of cards to be dealt from an arbitrarily shuffled deck of…
2000
The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from the same suit is:
- A.
3
- B.
8
- C.
9
- D.
12
Attempted by 40 students.
Show answer & explanation
Correct answer: C
Concept: The Pigeonhole Principle says that to FORCE some category to hold at least r items out of k categories, you must first exhaust the worst case of (r - 1) items sitting in every category, and only the very next item is guaranteed to push some category to r.
A standard deck has 4 suits, so there are k = 4 categories to distribute cards into.
Worst case: draw 2 cards from each of the 4 suits before any suit reaches 3 cards. This uses 2 × 4 = 8 cards, with no suit yet having 3 cards.
The 9th card drawn must belong to one of the 4 suits, and that suit already holds 2 cards, so it now holds 3 cards.
Hence the minimum number of cards that guarantees three cards from the same suit is 9.
Cross-check: With only 8 cards, the worst-case split of exactly 2 cards in each of the 4 suits is achievable, so 8 cards do not guarantee three cards of any suit — confirming that 9 is the smallest number for which the guarantee always holds.