Questions A and B

Question

A) How many ways can we select a 3-person committee from a group of 8 people.
B) How many ways can a President,
a Vice President, and a Secretary
be chosen from a group of 8 people?

Answer 1

A) choose 3 people from a group of 8 people

    8!/3!(8-3)!=8!/3!5!=8*7*6/3*2=8*7=56 ways you can choose 3 people from a group of 8 people

B) from 8 people 1 can be President

    from 7 people 1 can be Vice-President

    from 6 people 1 can be Secretary

   8*7*6=336 ways to choose a P, a VP, and a S

   look at it this way, you have 56 groups of 3 people

   call the people A, B, and C

   you want President, Vice-P, and Secretary

   ABC can be P, VP, and S

   ACB can be P, VP, and S

   BAC can be...

   BCA

   CAB

   CBA

   in each group of 3 people you can arrange them in 6 different ways; choose from 3 for P, 2 for VP, and 1 for S

   3*2*1=6

   there are 56 groups and each group can be arranged in 6 different ways

   56*6=336 ways you can choose a P, a VP, and a S

  

Answer 2

a) ₈C₃ = 8! /(3!(8-3)!)  =8!/6(5!)  = 8*7*6/6= 56

 

b) ₈P_{3 =  8!/(8 - 3)! = 8!/5! = 8*7*6 =} 336.