Discrete Mathematics: solve the following problems. justify your answers.

by

Discrete Mathematics

Solve the following problems. Justify your answers.

  1. How many binary strings of length n do not contain 10?
  2. Find a recurrence relation for the number of length-n ternary strings (strings using values 0, 1, 2) without two consecutive 0s. What are the initial values?
  3. In how many ways can you cover a 2 x n chessboard by 2 x 1 dominoes (placed horizontally or vertically)? Find a recurrence relation and initial values.
  4. How many subsets are there of the set {1, 2, . . . , n} that DO NOT contain three consecutive integers?Find a recurrence relation and initial values.
  5. Which is larger 2100 or F100?
  6. Use induction to prove
    F2 + F4 + F6 + ··· + F2n = F2n+1 – 1.
  7. Prove (without using induction)
    F 2 n+1 – F n 2= Fn−1Fn+2.