Question

Six students are to be seated in a row of 6 chairs. if three of thses students must be seated together, how many ways could this be accomplished?

Answer 1

students : A B C D E F
3 of them (A, B, & C) must be seated together

options::
A B C  D E F ..... D E F  A B C ..... D  A B C  E F ..... D E  A B C  F
A B C  D F E ..... D F E  A B C ..... D  A B C  F E ..... D F  A B C  E
A B C  E F D ..... E F D  A B C ..... E  A B C  F D ..... E F  A B C  D
A B C  E D F ..... E D F  A B C ..... E  A B C  D F ..... E D  A B C  F
A B C  F E D ..... F E D  A B C ..... F  A B C  E D ..... F E  A B C  D
A B C  F D E ..... F D E  A B C ..... F  A B C  D E ..... F D  A B C  E

24 options

Answer 2

For an instance, A,B,C,D,E,F represent the students, and ABC must be seated together.
Use factorial of 4 (4!). Why 4? Because we have 4 different positions. ABC together, and D, E, F who can be seated separately.
4! = 4 × 3 × 2 × 1
4! = 24

After that, multiply the result of the factorial above by 6. Why 6? Because ABC could have different order: ABC, ACB, BAC, BCA, CAB, CBA, as long as they are together. Therefore,
number of ways = 24 × 6
number of ways = 144

There are 144 ways