Permutations or Combinations?

Question

1. A Poker hand consists of 5 cards dealt at random from a deck of 52 cards. There are 2 598 960 possible hands (52C5). How many of those hands include exactly two Queens? Hint: You need two of the four queens, and three cards from those that are not queens.

2. A class of 8 students are giving presentations. The teacher needs to put them in order. Arun does not want to go first or last. In how many different ways could the order be arranged to keep Arun happy?

Answer 1

 

 There are (4 choose 2)= 4*3/2 = 6 ways to select the 2 queens

 

 Now we need 3 cards that are NOT queens.

 There are 48 such cards from which to pick 3 where order is irrelevant.

 

  (48 choose 3) = 48*47*46/6 = 17296

 

17296 * 6 = 103776

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There are 6 positions that Arun can be assigned,
namely positions 2,3,4,5,6,7;

Once that selection is made, there are only 5 positions left, besides the first

THere are 7 students that can fill the 1st position
There are 6 students to fill the one of the 5 positions left.
There are 5 students to fill one of the 4 positions left
There are 4 students to fill one of the 3 positions left
There are 3 students to fill one of the 2 positions left
There are 2 students to fill the last of position
The remaining student will be last (position 8)

6*7*6*5*4*3*2 = 30240