What is the number of ways of choosing 4 cards from a pack of 52 playi
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18 Apr 2019, 03:10
Question: What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these
(i) four cards are of the same suit,
(ii) four cards belong to four different suits
Bookish Solution:
There will be as many ways of choosing 4 cards from 52 cards as there are combinations of 52 different things, taken 4 at a time. Therefore
The required number of ways = 52C4= 270725
(i) There are four suits: diamond, club, spade, heart and there are 13 cards of each suit. Therefore, there are 13C4 ways of choosing 4 diamonds. Similarly, there are
13C4 ways of choosing 4 clubs, 13C4 ways of choosing 4 spades and 13C4 ways of choosing 4 hearts.
Therefore, The required number of ways = 13C4 + 13C4 + 13C4 + 13C4 = 2860
(ii) There are 13 cards in each suit. Therefore, there are 13C1 ways of choosing 1 card from 13 cards of diamond, 13C1 ways of choosing 1 card from 13 cards of hearts, 13C1 ways of choosing 1 card from 13 cards of clubs, 13C1 ways of choosing 1 card from 13 cards of spades. Hence, by multiplication principle, the required number of ways
= 13C1 × 13C1 × 13C1× 13C1 = 13^4 = 28561
My doubt is for {ii} part:
I subtracted the number of arrangements in which all the 04 cards are of same suit from all number of ways of choosing 4 cards from a pack of 52 playing cards
i.e. # of ways of choosing 04 cards belonging to four different suits =
(# of ways of choosing 4 cards from a pack of 52 playing cards) - (# of ways of choosing 04 cards of the same suit)
= 270725 - 28561 = 242164, which is not the correct answer.
Kindly guide me where I am committing mistake or where is the concept gap.