Five cards are chosen at random from an ordinary deck of 52 cards. In how many ways is it possible to choose exactly three clubs

Question

Five cards are chosen at random from an ordinary deck of 52 cards 

Answer 1

The number of ways to choose three clubs is C_13,3 or 13 clubs in a deck choose 3. The number of ways to choose two non-clubs is C_39,2. Note the subscripts sum to 52,5 - denoting 5 cards in a standard 52 card deck. The number of ways to choose 5 cards in a 52 card deck is C_52,5. So the answer is C_13,3C_39,2/C_52,5

 

C_13,3 = 13!/(10!*3!) = 13*12*11/6 = 13*2*11 = 286

 

C_39,2 = 39*38/2  = 741

 

Figure out C_52,5 and do the math and you will have the answer.