Try It Now

1. The judiciary committee at a college is made up of three faculty members and four students. If ten faculty members and 25 students have been nominated for the committee, how many judiciary committees could be formed at this point?
   
   
2. Suppose that a single character is stored in a computer using eight bits.
   
   1. How many bit patterns have exactly three 1s?
   2. How many bit patterns have at least two 1s?
      
      Hint. Think of the set of positions that contain a 1 to turn this is into a question about sets.
      
      
3. How many subsets of {1,2, 3, . . . , 10}contain at least seven elements?
   
   
5. The image below shows a 6 by 6 grid and an example of a lattice path that could be taken from (0, 0) to (6, 6), which is a path taken by traveling along grid lines going only to the right and up. How many different lattice paths are there of this type? Generalize to the case of lattice paths from (0, 0) to (m, n) for any nonnegative integers m and n.
   
   
   Figure 2.4.12: A lattice path
   
   Hint. Think of each path as a sequence of instructions to go right (R) and up (U).
   
   
7. A poker game is played with 52 cards. At the start of a game, each player gets five of the cards. The order in which cards are dealt doesn't matter.
   1. How many "hands" of five cards are possible?
   2. If there are four people playing, how many initial five-card "hands" are possible, taking into account all players and their positions at the table? Position with respect to the dealer does matter.
      
      
9. How many five-card poker hands using 52 cards contain exactly two aces?
   
   
11. A class of twelve computer science students are to be divided into three groups of 3, 4, and 5 students to work on a project. How many ways can this be done if every student is to be in exactly one group?
    
    
13. There are ten points, P₁, P₂, . . . , P₁₀ on a plane, no three on the same line.
    
    1. How many lines are determined by the points?
    2. How many triangles are determined by the points?
       
       
15. Use the binomial theorem to prove that if A is a finite set, then $\left |P (A) \right | = 2^{\left |A \right |}$
    
    
17. Use the binomial theorem to calculate 9998³.
    
    Hint. 9998 = 10000 − 2

Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
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