Practice Problems Section 5.6 and Pascal’s Triangle

These practice problems are not in the text book:

  1. Generate the first 11 rows of Pascals Triangle (stop when you’ve reached C(10,10)).
  2. Use your answer to number one to find C(8,6) (do not use the formula \binom{8}{6}).
  3. Check that the sum of all of the entries in the last row in your triangle is 2^10. What are you counting when you add the entries in a row of Pascal’s Triangle? (Solution: When you add all of the entries of the last row in your triangle you’re counting all of the subsets of the set {1,2,3,…,10}.)
  4. Verify that the number of all subsets of the set {1,2,3,4} is 2^4 either by using Pascal’s Triangle or by writing down all possible subsets.
  5. Use multiplication to count all of the subsets of the set {1,2,3,4}. (Hint: for each number between 1 and 4, there are two options: either the number is In the subset or it is Out. For each subset, write down a corresponding sequence of I‘s and O‘s. Now the problem looks just like counting all possible outcomes when we toss a coin 4 times.)
  6. Verify that the alternating sum of the last row from your triangle is 0. (The alternating sum means that we alternate between adding and subtracting. For example, the alternating sum of the 4-th row in Pascal’s Triangle is 1-3+3-1.)

The follow problems come from Section 5.6 of your text book:

  • 1,2,3,4,9,11,13,15 (For 9-15 odd, shortest possible route means that only South and East steps are allowed.)
  • Solutions:
    • 2 Part a: 2^9, Part b: C(9,2)
    • 4 Part a: C(6,0)+C(6,1)+C(6,2), Part b: 2^6-22 (Hint: Think about our solution to number 1 in Quiz 7),

 

Here are selected solutions for problems not in your text book:

Download (PDF, 186KB)

Here’s a link to download: Selected solutions