Question

what is the number of ways of choosing 6 cards from a pack of 52 playing cards ? In how many of these three are red and three are black​

Answer 1

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