39 votes 39 votes The minimum number of cards to be dealt from an arbitrarily shuffled deck of $52$ cards to guarantee that three cards are from same suit is $3$ $8$ $9$ $12$ Combinatory gatecse-2000 easy pigeonhole-principle combinatory + – Kathleen asked Sep 14, 2014 edited Jun 7, 2018 by Milicevic3306 Kathleen 9.9k views answer comment Share Follow See all 7 Comments See all 7 7 Comments reply Show 4 previous comments Lakshman Bhaiya commented Nov 8, 2019 reply Follow Share This is a clear picture of 52 cards. 8 votes 8 votes KUSHAGRA गुप्ता commented Jan 13, 2020 reply Follow Share The minimum # of pigeons which assures atleast K+1 pigeons in some pigeon hole=Kn+1 $Here\ K+1=3\ \&\ n=4(suits)$ $\therefore 2\times 4+1=9$ 0 votes 0 votes Akash 15 commented Dec 19, 2023 reply Follow Share In the deck of $52$ cards there are $4$ suits $[Heart\;\;Diamond\;\;Spade\;\;Club]$. So, No. of holes = $4$ Now in each suit we pick $2$ cards. So, we picked $8$ cards and we will pick another card from same suit to guarantee that $3$ cards are from same suit. $\therefore$ Min. no of cards = $8+1=9$ 0 votes 0 votes Please log in or register to add a comment.
Best answer 42 votes 42 votes There are $4$ sets of cards. So, up till $8$ cards there is a chance that no more than $2$ cards are from a given set. But, once we pick the $9$$^{th}$one, it should make $3$ cards from any one of the sets. So, $(C)$ is the answer. gatecse answered Sep 14, 2014 edited Jun 8, 2018 by Milicevic3306 gatecse comment Share Follow See all 3 Comments See all 3 3 Comments reply indrajeet commented Aug 2, 2016 reply Follow Share may be in first 3 attempt we get same(suits) card and question is asking about minimum. then A (3) is best choice 0 votes 0 votes Prashant. commented Aug 2, 2016 reply Follow Share minimum in worst case. 5 votes 5 votes Sourav Basu commented Nov 3, 2017 reply Follow Share But it will not guarantee that three cards are from same suit. Only Picking 9 or more ( 9 or10 or ......or 52) will guarantee that three cards are from same suit. Minimum of( 9 ,10, 11...52)= 9, So, 9 is the answer(minimum number of cards to be dealt to guarantee that three cards are from same suit) 4 votes 4 votes Please log in or register to add a comment.
23 votes 23 votes As suggested above also apply pigeon hole, 4 holes (suits) n pigeons(no of cards to be drawn) floor [(n-1)/p] +1=3 floor[(n-1)/4] =2 (n-1)/4 >= 2 n>=9 minimum 9 cards must be picked Anurag_s answered Dec 29, 2015 Anurag_s comment Share Follow See all 0 reply Please log in or register to add a comment.
9 votes 9 votes Any corrections or questions to my solution are welcome Suneel Padala answered Dec 14, 2018 Suneel Padala comment Share Follow See 1 comment See all 1 1 comment reply Sona Barman commented Apr 14, 2019 reply Follow Share I could not understandem logic behind to apply pigeonhole principle. But this technique help me lot. 0 votes 0 votes Please log in or register to add a comment.
5 votes 5 votes Can this question be solved by calculating expectation: Expectation of getting three cards from same deck-> E(x) = 13*(13/52) + 12*(12/51) + 11*(11/50) = 8.49 => 9 cards Hence 9 is the answer. Is it correct? shikharV answered Dec 10, 2015 shikharV comment Share Follow See 1 comment See all 1 1 comment reply Swati Rauniyar commented Oct 17, 2017 reply Follow Share @shikhar May you please explain your approach more? I find it's correct use of "Expectation". 0 votes 0 votes Please log in or register to add a comment.