Example 19 - What is number of ways of choosing 4 cards from - Combination

  1. Chapter 7 Class 11 Permutations and Combinations
  2. Serial order wise

Transcript

Example, 19 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these four cards are of the same suit, There are four suits i.e. diamond, spade, heart, club & 13 card of each suit Since, they are different cases, So, we add the number of ways The required number of ways choosing four cards of the same suit = 13C4 + 13C4 + 13C4 + 13C4 = 4 × 13C4 = 4 × 13!/(4!(13 − 4)) = 4 × 13!/(4! 9!) = 4 × (13 × 12 × 11 × 10 × 9!)/(4! × 3 × 2 × 1 × 9!) = 4 × (13 × 12 × 11 × 10)/(4 × 3 × 2 × 1) = 2860 ways Example, 19 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (ii) four cards belong to four different suits, Since, they are the same case, So, we multiply the number of ways Hence the required no of ways choosing four cards from each suit = 13C1 × 13C1 × 13C1 × 13C1 = (13C1)4 = (13!/1!(13 − 1)!)^4 = (13!/1!12!)^4 = ((13 × 12!)/12!)^4 = (13)4 = 13 × 13 × 13 × 13 = 28561 ways Example, 19 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (iii) are face cards, King Queen and Jack are face cards Number of face cards in One suit = 3 Total number of face cards = Number of face cards in 4 suits = 4 × 3 = 12 Hence, n = 12 Number of card to be selected = 4 So, r = 4 Required no of ways choosing face cards = 12C4 = 12!/4!(12 − 4)! = 12!/4!8! = (12 × 11 × 10 × 9 × 8!)/(4 × 3 × 2 × 1 × 8!) = (12 × 11 × 10 × 9 )/(4 × 3 × 2 × 1 ) = 495 ways Example, 19 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (iv) two are red cards and two are black cards, Since, they are the same case, So, we multiply the number of ways Total no of ways choosing 2 red & 2 black cards = 26C2 × 26C2 = (26C2)2 = (26!/(2! (26 − 2)!))^2 = (26!/(2! 24!))^2 = ((26 × 25 × 24!)/(2 × 1 × 24!))^2 = ((26 × 25)/(2 × 1))^2 = (13 × 25)2 = (325)2 = 105625 Example, 19 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (v) cards are of the same color? Since, choosing red OR black , they are different cases, So, we add the number of ways Total number of ways selecting four cards of same colour = 26C4 + 26C4 = 2(26C4) = 2 × 26!/4!(26 − 4)! = 2 × 26!/(4! 22!) = 2 × (26×25×24×23×22!)/(4×3×2×1×22!) = 2 × (26×25×24×23)/(4×3×2×1) = 29900

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