Permutations and combinations：練習問題

A student has 7 different books. In how many different ways can the student select 3 books to take on a holiday?

A group of 10 people, consisting of 6 men and 4 women, is to be seated in a single row of 10 chairs. Find the number of different ways they can be seated if all 4 women must sit together in a single block.

A committee of 5 people is to be chosen from a group of 7 men and 6 women. Find the number of different ways the committee can be formed if it must contain at least 3 women and at least 1 man.

Find the number of different ways in which all 8 letters of the word PARALLEL can be arranged in a straight line.

Find the number of different 7-digit numbers that can be formed using all the digits from the set \(\{1, 2, 2, 3, 4, 4, 4\}\) such that the number is greater than \(3\,000\,000\).

A sports team consists of 6 players from Year 12 and 6 players from Year 13. A starting lineup of 5 players must be chosen such that it contains at least one player from each year group. Calculate the number of different lineups possible.

A delegation of 6 people is to be chosen from 5 teachers and 6 students. Find the number of different ways to form the delegation if it must contain at least 2 teachers, at least 2 students, and a specific teacher and a specific student refuse to serve together.

Find the number of different ways the letters of the word STRESS can be arranged in a line if the three 'S's must be kept together as a single block.

A 4-digit code is to be formed using the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\).
(a) Find the number of different 4-digit codes that can be formed if no digit may be used more than once.
(b) Find how many of the codes in part (a) represent numbers that are greater than \(5000\) and are odd.

A sports club has 15 members, consisting of 8 men and 7 women. A team of 6 members is to be selected to represent the club in a competition.
(a) Find the number of different teams that can be formed if the team must contain at least 2 men and at least 2 women.
(b) Two of the men, Alan and Ben, refuse to be in the team together. Find the number of different teams of 6 that can be formed if this restriction is applied and the team must also contain exactly 3 women.