Question

There is a workshop of 20 people. During a section of the workshop, the people are divided randomly into groups of 4. There are six tables for six people.

Each group was randomly assigned to sit at one of the six tables.

a) How many ways can four people be picked to be in the first group formed?

b) After the first group is formed, how many ways can four people be picked to be in the second group formed?

c) Given that the order in which the groups were formed doesn't matter, how many different ways could the 20 people be divided into the groups?

d) After the groups are formed, how many ways can the groups be assigned to the six tables?

e) Once assigned to a table, how many distinguishable ways are there for the group members to select their seals.

Answer 1

Part (a)

4 out of 20 can be selected in ²⁰C₄ ways = (20 x 19 x 18 x 17)/24 = 4845 ANSWER

Part (b)

Once first group of 4 is formed, the second group of 4 can be selected out of the remaining 16 people in ¹⁶C₄ ways = (16 x 15 x 14 x 13)/24 = 1820 ANSWER

Part (c)

Continuing the logic as above further, the third group can be selected in ¹²C₄ ways, fourth group in ⁸C₄ ways and the fifth group in ⁴C₄ ways. Thus, total number of selections would be:

(²⁰C₄)( ¹⁶C₄)( ¹²C₄)( ⁸C₄)( ⁴C₄) = (20!)/{(4!)}⁵. Since order is not important, this must be divided by 5! Because 5 groups can be arranged in 5! ways. Thus, final answer is:

(20!)/[{(4!)}⁵ x (5!)] ANSWER

Part (d)

There are 5 groups and 6 tables. Group 1 can occupy any one of the 6 tables. Then Group 2 can occupy any one of the remaining 5 tables, Group 3 can occupy any one of the remaining 4 tables, Group 4 can occupy any one of the remaining 3 tables, and finally Group 5 can occupy any one of the remaining 2 tables.

Thus, total number of arrangements = (6 x 5 x 4 x 3 x 2) = 720 ANSWER