Use what you have learned about combinations to work out the following problems. Permutations and other counting rules we've covered may also be required. 1. For a magic trick, you ask a friend to draw three cards from a standard deck of 52 cards. How many possible sets of cards might she have chosen? 2. For the same trick, you insist that your friend keep replacing her first draw until she draws a card that isn't a spade. She can choose any cards for her other two cards. How many possible sets of cards might she end up with? (Caution: choosing 5%, 64, 34 in that order, is not different from choosing 64, 5%, 34 in that order. You do not need to take into account that some sets will be more likely to occur than others.) 3. How many 5-digit numbers contain exactly two zeroes? (We insist that the number contain exactly 5 digits.) 4. Sandeep, Hee, Sara, and Mohammad play euchre with a standard deck consisting of 24 cards (A, K, Q, J, 10, and 9 from each of the four suits of a regular deck of playing cards). In how many ways can the deck be dealt so that each player receives 5 cards, with 4 cards left in the middle, one of which is turned face-up? The order of the 3 cards that are left face-down in the middle does not matter, but who receives a particular set of 5 cards (for example, Sara or Sandeep) does matter. 5. An ice cream shop has 10 flavours of ice cream and 7 toppings. Their mega-sundae consists of your choice of any 3 flavours of ice cream and any 4 toppings. (A customer must choose exactly three different flavours of ice cream and four different toppings.) How many different mega-sundaes are there