17 Among the 2589960 possible 5card poker hands from a stand

17. Among the 2,589,960 possible 5-card poker hands from a standard 52- card deck, how many contain at least one card that is not a club (compliment of \"all club\") ?

Solution

Total number of possibilities in a 5 card poker from a standard 52 card deck, N[total ways] = ⁵²C₅ = 2,598,960

Total number of clubs in the pack of 52 cards = 13

So number of cards which are not clubs = 52 - 13 = 39

N[at least one club] = N[total ways] - N\'[at least one club]

No of ways of choosing such that its at least one club, N\'[at least one club] = ³⁹C₅ = 575757

So no of ways of choosing at least one card that is not a club = ⁵²C₅ - ³⁹C₅ = 2,023,203