Homework Assignment Eight

You may collaborate on 3, 6, 12.

Citations in [ ] indicate problem numbers in Tucker, 6th edition.

  1. Show the correctness of this binomial identity using block-walking [#11 in 5.5]: ∑k=0k=mC(m,k)*C(n,r+k)=C(m+n,m+r)
  2. Prove that C(n,1)+3C(n,3)+5C(n,5)+ ... = 2C(n,2)+4C(n,4)+6C(n,6)... What is the value of each side? [5.5.22]
  3. Give a combinatorial argument to evaluate this sum, for n<=m: C(n,0)*C(m,0)+C(n,1)*C(m,1)+...+C(n,k)*C(m,k)+...C(n,n)*C(m,n) [5.5.26]
  4. Find the coefficient of x**24 in (x + x**2 + x**3 + x**4 + x**5)**8. [6.2.8]
  5. Use generating functions to model the number of different selections of r hot dogs when there are five types of hot dogs. [6.1.6]
  6. Show that the generating function for the number of integer solutions to a+b+c+d = r,0<=a<=b<=c<=d, is (1+x+x**2+x**3+...)(1+x**2+x**4+..)(1+x**3+x**6+..)(1+x**4+x**8+..) [6.1.22]
  7. Find the coefficient of x**r in (x**5 + x**6 + x**7+...)**7 [6.2.2]
  8. Find a generating function for the number of integers between 0 and 999,999 whose digits sum to r. [6.1.16]
  9. Use a generating function for modeling the number of distributions of 20 identical balls into 5 distinct boxes if each box has between 2 and 7 balls. [6.1.18]
  10. How many ways are there to paint 10 identical rooms in a hotel with 5 colors if at most 3 rooms can be painted green, at most three can be painted blue, at most three can be painted red, and no constraints on the other two colors, black and white? [6.2.20]
  11. How many ways are there to get a sum of 25 when 10 distinct dice are rolled? [6.2.22]
  12. Show that (1 - x - x**2 - x**3 - x**4 - x**5 - x**6)-1 is the generating function for the number of ways a sum of r can occur if a die is rolled any number of times. [6.2.30]
Due: In class, Thursday 10/30

-- RobbieMoll - 01 Apr 2009

Topic revision: r7 - 2014-10-27 - RobbieMoll
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