Counting with Repeated Objects 451- The Old Town Softball League has 16 teams arranged in four groups of four teams
each. How many different ways can these groups be made up? - How many ways can a committee be selected consisting of two Independents, two
Republicans, and two Democrats if the choices are made from seven Independents,
nine Republicans, and eight Democrats? - A team of 11 players is to be chosen from a group of 15 candidates.
(a) How many different teams can be chosen?
(b) How many teams can be chosen if one player is designated captain and must play
on the team?- How many seven-digit sequences can be formed using the symbols J0, 1, 2, 3)?
- How many four-person teams can be formed from three men and five women if at least
one man and at least one woman are on each team? - (a) Which permutation of {1, 2, 3, 4, 5) follows 3-1-5-2-4 in the lexicographical or-
dering of the permutations of five elements?
(b) Repeat the question for 4-6-1-3-7-5-2 as a permutation of [1, 2 ... , 7}. - (a) Construct the permutation numbered 39 in the dictionary ordering of the permuta-
tions of the letters {1, 2, 3, 4, 5). Remember this is actually the 40th permutation,
since the numbering of permutations starts with 0.
(b) Construct the permutation numbered 387 in the permutations of {1, 2, 3, 4, 5, 6).
(c) Construct the permutation numbered 3764 in the permutations of { 1, 2, 3, 4, 5,
6,7}.
(d) Construct the permutation numbered 27,459 in the permutations of { 1, 2, 3, 4, 5,
6,7,81. - What is the 311th permutation of {1, 2, 3, 4, 5, 6,^71 relative to the lexicographical
ordering? (Remember that the 31 1th is numbered 310, since the first element is num-
bered 0). What is the 2374th? - A football team of 11 players is to be selected from a set of^15 players. Five players
in this set can only play in the backfield, eight can only play on the line, and two can
play either in the backfield or on the line. A team has seven players on the line and
four players in the backfield. How many different teams can be selected? - A classroom has two rows of eight seats. There are 14 students in the class. Five
students always sit in the front row, and four always sit in the back row. In how many
ways can the students be seated? - How many ways are there to roll 10 dice so that all six different faces show?
Counting with Repeated Objects
Although the word abracadabra has 1 letters, there are not 11! permutations that result
in different words. For example, the first two occurrences of a can be interchanged, and no
different word would be apparent.
In the word abracadabra the letters a, b, and r occur more than once. If we permute the
letters of this word, then it is impossible, for example, to recognize two anagrams as being
different when they simply interchange different pairs of occurrences of a repeated letter.
A generalization of results about permutations and combinations is needed when some