Question

There are 15 members on the board of directors for a Fortune 500 company. If they must select a chairperson, a first vice chairperson, a second vice chairperson, and a secretary.

(a) How many different ways the officers can be selected? Show work.

(b) Please describe the method used and the reason why it is appropriate for answering the question.

A researcher wants to conduct a clinical trial on a new medicine for a rare disease. She plans to randomly choose 4 testers from 20 eligible patients.

(a) How many different ways can the testers be chosen? Show Work

(b) Please describe the method used and the reason why it is appropriate for answering the question

Answer 1

4 persons needs to be selected out of 15

a) ways the officers can be selected = 15P4 ( without repetition and order does matter )

ways the officers can be selected = 15_fact /(15-4)_fact = 15_fact / 11_fact = 15*14*13*12 = 32760 ways

b) Reason : For first position we have 15 choices, for second we have 14, for 3rd we have 13 choices and at 4th we have 12 choices. So, number of ways the officers can be selected = 15*14*13*12 = 32760

4 testers from 20 eligible patients

a) ways the patient can be selected = 20C4 ( without repetition and order does not matter )

ways the patient can be selected = 20C4 = 20_fact / ( 4_fact*16_fact ) = 20*19*18*17 / 4*3*2*1 = 4845 ways

b) Reason : For first position we have 20 choices, for second we have 19, for 3rd we have 18 choices and at 4th we have 17 choices and each combination can have 4_fact = 4*3*2*1 = 24 ways of arranging. Therefore, number of ways the officers can be selected = 20*19*18*17 / 4*3*2*1 = 4845 ways