Induction Proof

If n >= 1, the number of strings using the digits 0,1, and 2 with no two consecutive places holding the same digit, is 3x2^n-1. For example, there are 12 such strings of length three: 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210, and 212. Prove this claim by induction on the length of the strings. Is the formula tr ...continues

Induction Problem

Attached in word document Numbers of the form tn = n (n+1) / 2 are called triangular numbers, because marbles arranged in an equilateral triangle, n on a side, will total... (see attachment)

Error Correcting Code

If no two strings in a code differ in fewer than three positions, the we can actually correct a single error, by finding the unique string in the code that differs from the received string in only one position. It turns out that there is a code of 7-bit strings that corrects single errors and contains 16 strings. Find such a c ...continues

Recursive definition

We can define sorted lists of integers as follows: BASIS - A list consisting of a single integer is sorted. INDUCTION - If L is a sorted list in which the last element is a and if b >= a, then L followed by b is a sorted list. Prove that this recursive definition of "sorted list" is equivalent to our original, nonrecurs ...continues

Induction by recursion

Define recursively the set of even-parity strings, by induction on the length of the string. Hint: It helps to define two concepts simultaneously, both the even-parity and odd-parity strings.

Discrete e.1 Send SOLUTION as attachment.

Let S = {1,2,3,4,5,6,7,8}. Determine: (a) The number of subsets of S (b) The number of subsets of S with at most four elements ... (see attachment for rest)

Discrete

Let Pn be the product of the first n odd numbers... (see attachment)

Discrete e.3 Send SOLUTION as attachment

3. At the Clone Zone School a class of 10 identical boys and 8 identical girls go to a cafeteria... (see attachment)

Provide a combinatorial proof

4. Provide a combinatorial proof: For postive integers N.... (see attachment) Please send response as a .doc attachment.

Find an explicit formula

5. Let TN be the number of ways a 2 X N rectangle can be tiled with 2 X 2 squares... (see attachment)