Prove that a parity check bit can detect an error in a string if and only if the string contains an odd number of errors.

A decimal string is a string over the alphabet of the 10 characters: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. How many decimal strings of length 6 are there that consist of 6 different characters in decreasing order? How many decimal strings of length 6 are there that have at least one letter OR at least one 0?

Find the number of all bit strings of length 5.

How many bit strings of length 10 contain either five con- secutive Os or five consecutive 1s? How many bit strings of length eight contain either three consecutive 0s or four consecutive 1s?

A small island country with less than 20,000 cars has decided to reissue the license plates for its cars and has decided on the following rules for assigning numbers and letters to the license plates. Each plate will have 7 character-positions that each must be filled. The leftmost character position will hold a letter from the set {D, E, H, G, X}. The next three positions will hold integers from the set {1, 9} but none of the integers can be repeated. The rightmost three positions must be entirely from the set {I, V, X} or entirely from the set {P, Q, R}, with repetition allowed in either case. For example, a valid plate would be G 127 XXI and an invalid plate would be G 337 PQX (two rules violated). What is the maximum number of different license plates following the rules? O 270,000 O 3,645,000 194,400 136,080