UGC NET CSE | June 2019 | Part 2 | Question: 7

Question

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

• A. $9$
• B. $13$
• C. $17$
• D. $42$

Comments

Answer will be $42$, with pegion hole principle

---

Rosen example..

---

how would it be applied? please show.

Answer 1

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.

Comments

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.

---

I did not get .plz can u explain it in more words?

---

The worst case, we may select all the clubs,
diamonds, and spades (39 cards) BEFORE ANY HEARTS. (i.e 13 clubs+ 13 diamonds +13 spades, which is equal to 39 cards)

So, to guarantee that at least three hearts are selected,
39+3=42 cards should be selected.

Answer 2

To guarantee the Event at first you have to prevent the event

to prevent the event you have to take at most 2 hearts(<=2) ,13 ,13 , 13 each  total 13*3+2=41 to prevent the event.

and to guarantee the event you have to take 41+1=42 card.