Homework Assignment Nine

You may collaborate on numbers 4,5, 8. Notation in brackets are references to problems in Tucker, 5th edition.

1. Find the coefficient of x¹⁸ in (1+x³+x⁶+x⁹+...)⁷[6.2.14]
2. In the US there are 50 states, each with 2 senators. How many different committees of 40 senators can be formed if the two senators from the same state are considered to be identical? [6.2.24]
3. How many ways are there to divide 5 pears, 5 apples, 5 doughnuts, 5 lollipops, 5 chocolate cats, and 5 candy rocks into two (unordered) piles of 15 objects each? [6.2.26]
4. How many r-digit ternary sequences are there in which a) no digit occurs exactly twice; and b) 0 and 1 each occurs an even number of times. [6.4.10]
5. Find a recurrence relation for the number of ways to go N miles by foot walking at 3 mph, or jogging 5 mph, or running 10 mph; at the end of each hour a choice is made about how to go during the next hour. [7.1.4]
6. Find a recurrence relation for the number of n-digit binary sequences with a) no pair of consecutive 1s; b) no pair of consecutive 1s, but for ternary sequences; and c) no pair of consecutive 0s and 1s, for ternary sequences. [7.1.6]
7. Show that s_N = C(N+1,0)+ C(N,1) + C(N-1, 2)... satisfies the Fibonacci relation.[7.1.8]
8. Find and solve a recurrence for the number of n-digit ternary sequences with no consecutive digits being equal. [7.3.2]

-- RobbieMoll - 08 April 2009