Permutations and combinations : แบบฝึกหัด

A bakery offers 6 distinct types of cookies and 4 distinct types of muffins. A customer wants to choose a selection consisting of exactly 2 cookies and 1 muffin. How many different selections are possible?

A photographer needs to arrange 7 students, consisting of 4 boys and 3 girls, in a straight line for a group photo. In how many distinct ways can they be arranged if all three girls must always sit together?

There are 4 distinct boys and 4 distinct girls to be arranged in a straight line. The arrangement must satisfy two conditions: (i) boys and girls must alternate, and (ii) a particular boy, \(B_1\), and a particular girl, \(G_1\), must not sit next to each other.
Find the total number of possible arrangements.

A bakery offers 5 different types of bread and 4 different types of cake. A customer wants to select 2 distinct types of bread and 1 type of cake. How many different selections are possible?

A team of 5 people is to be selected from a group consisting of 8 men and 6 women. Find the number of different teams that can be formed if the team must contain at least 4 men.

How many different ways can 5 distinct books be arranged on a shelf if a particular book must be at the first position?

A box contains 8 red balls and 6 blue balls. If 4 balls are drawn randomly, how many ways are there to draw at most 1 blue ball?

In how many ways can 10 different runners be positioned at the starting blocks if runner A and runner B must be in adjacent lanes, and runner C and runner D must not be in adjacent lanes?

A company team consists of 5 men and 4 women. A committee of 4 people is to be formed from this team.

(a) Find the total number of different committees that can be formed from the team.

(b) Find the number of different committees that can be formed if the committee must consist of exactly 2 men.

(c) Suppose a committee is formed consisting of 2 men and 2 women (as found in (b)). If these 4 committee members are arranged in a straight line for a photograph, find the number of different arrangements such that the 2 men must sit next to each other.

Eight distinct letters, P, Q, R, S, T, U, V, and W, are to be arranged in a row.
(a) In how many ways can the letters be arranged if P and Q must be together?
(b) In how many ways can the letters be arranged if P and Q must not be together?