Question

what is the number of choosing 6 cards from a pack pf 52 cards if six cards belong to four different suits?​

Answer 1

Answer:

there are four suits in a deck of 52 cards. clubs , spades, diamonds and hearts.

case 1. we want all 4 from the same suit. i.e either of hearts or of clubs or of diamonds or of spades.

no of ways of piking 4 cards from clubs= no of ways of picking 4 cards from spades= no of ways of piking 4 cards from hearts= no of ways of piking 4 cards from diamonds.

no of ways of piking 4 cards from say hearts are 13C4.

so total no of ways of picking 4 cards where all are of same suit= 13C4 +13C4+ 13C4+ 13C4 = 4*(13C4).

case 2 we want all 4 from different suits that is possible in only one way . if one card is from hearts one is from clubs one is from diamonds , one is from spades =

so total no of ways = 13C1 *13C1* 13C1* 13C1 = (13C1)^4 = 13^4

Answer 2

The number of ways to choose 4 cards from 52 is 52C4 = (52 x 51 x 50 x 49)/(4 x 3 x 2) = 13 x 17 x 25 x 49 = 270,725.

(i) The number of ways the 4 cards are from the same suit is 13C4 x 4 = (13 x 12 x 11 x 10)/(4 x 3 x 2) x 4 = 13 x 11 x 5 x 4 = 2,860 (Note: 13C4 is the number of ways of choosing 4 cards from one suit of 13 cards and we multiply by 4 since there are 4 suits.)

(2) The number of ways the 4 cards are from the 4 different suits is 13C1 x 13C1 x 13C1 x 13C1 = 13 x 13 x 13 x 13 = 28,561.