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How many cards must be selected from a standard deck of $52$ cards to guarantee that at least three hearts are present among them?

1. $9$
2. $13$
3. $17$
4. $42$

edited | 280 views
+1
Answer will be $42$, with pegion hole principle

+1 vote

In this the worst case will happen as follows :-

$1^{st}$ $13*3 = 39$ cards that we pick up turns out to be any one of daimond or spades or clubs but not hearts i.e. Now when we pick the next $3$ cards they are guaranteed to be hearts.($\because$ we have already picked cards from all other suits)

$\therefore$ We need to pick $13+13+13+3= 42$ cards in order to guarantee that there are atleast $3$ cards of hearts in the selected cards.

So option $4$ is the correct option.

by Boss (21.5k points)
edited by
+3
Thats not exactly the worst case. In worst case the first 39 will not have any heart -- but they can be in any order. To be more precise, by having only $2$ hearts, we have only $39$ other cards and these $41$ can be in any order.