Permutations and combinations：練習問題

Calculate the number of different ways all the letters of the word VECTOR can be arranged in a line.

How many different 4-digit numbers can be formed using the digits 1, 3, 5, 7, 8, and 9, if each digit can be used only once and the number must be even?

A committee of 5 members is to be chosen from a group of 6 men and 4 women. Find the number of different ways the committee can be formed if it must contain at least 3 women.

Find the number of different 5-letter arrangements that can be formed using all the letters of the word ORBIT if the first letter must be a vowel.

Four men and three women are to stand in a straight line for a photograph. Calculate the number of different ways they can be arranged if no two women are to stand next to each other.

A student has 7 different posters to display. Find the number of ways that 3 of these posters can be selected and arranged in a row on a wall.

A password is to be formed using 4 characters from the set \(\{A, B, C, D, E, 1, 2, 3\}\). No character can be repeated. Find the number of different passwords that can be formed if the password must start with a letter and end with a digit.

Consider the letters of the word CHAMPION. Find the number of different 8-letter arrangements that can be made if the vowels (A, I, O) must not be next to each other.

A sports club needs to select its leadership for the upcoming season from a group of 12 eligible members.
(a) Find the number of ways to choose a chairperson and a secretary.
(b) Find the number of ways to choose a committee of 3 members from the remaining 10 members.

There are 12 points marked in a plane. 5 of these points lie on a single straight line \(L\), and the other 7 points are scattered such that no three points (including those on \(L\)) are collinear. Find:
(a) The number of different straight lines that can be formed by joining any two points.
(b) The number of different triangles that can be formed using these points as vertices.