Question

There are 5 classes of Form 6 in a secondary school. To form a task group of 20 members, 4 representatives are nominated by each class. From the task group, 5 members are randomly selected. Find the number of ways to select the 5 members if they are nominated by
(a) five different classes;
(b) four different classes;
(c) three different classes.

Answer 1

4 representatives are selected from each of the 5 classes.

a)number of ways to select the 5 members from the group of 20, if they are nominated by 5 different classes

[1 member can be selected from 4 members of same class in 4C1 ways for each such cases another member can be selected from 4 members of another class in 4C1 ways and so on.]

b)number of ways to select the 5 members from the group of 20, if they are nominated by 4 different classes

[4 different classes can be selected from 5 classes 5C4 ways for each such cases,2 members can be selected from a class in 4C2 ways and that class can be selected from 4 classes in 4C1 ways, for each such cases rest 3 members can be selected from rest 3 classes in 43 ways ]

c)number of ways to select the 5 members from the group of 20, if they are nominated by 3 different classes

[first 3 classes are choosen from 5. Now, either 3 can be choosen from same class and the rest 2 from other 2 different classes, or, 2 from a class and 2 from another class and 1 from the rest class. For the first case, 1 class is selected from 3 in 3C1 ways from which 3 members get selected in 4C3 ways, for each such cases rest 2 members can be selected from rest 2 classes in 42 ways. For the other case, 2 classes are selected from 3 in 3C2 ways from each of which 2 members get selected in 4C2*4C2 ways, for each such cases rest 1 member can be selected from rest class in 4 ways.]