A question asks what activities students like out of a list: studying, internet surfing, video games,?

basketball, billiards, walking, tree planting. A student can pick no activity, 1 activity, or more than one activity. How many different ways could a student answer the question?

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• Answerer 1

  Actually, this problem calls for factorials.
  You know that there are seven possibilities.
  when the students pick 0 likes they can only do that one way and the same for if they say they like all 7. So for 0 picks you have one way to choose and for 7 picks you have 1 way to choose all seven.
  
  Choose 0 =1 way
  Choose 7 = 1 way
  Choose 1 = 7/(1!) = 7/1 = 7 ways (Seven possibilities to choose only one)
  Choose 2 = (7x6)/(2!) = 7x6/(2x1) = 42/2 = 21 ways
  Choose 3 = (7x6x5)/(3!) = (7x6x5)/(3X2X1) = 210/(3X2X1) = 36 ways
  Choose 4 = (7x6x5x4)/(4!) = (7x6x5x4)/(4x3X2X1) = 840/24 = 35 ways
  Choose 5 = (7x6x5x4x3)/(5!) = (7x6x5x4x3)/(5x4x3X2X1) = 2520/120 = 21 ways
  Choose 6 = (7x6x5x4x3x2)/(6!) = (7x6x5x4x3x2)/(6x5x4x3X2X1) = 5040/720 = 7 ways
  
  Add the ways up and you get 129 ways.
  
  To understand this lets look at choose 3:
  you can't choose the same thing more than once so there are 7 possibilities for the first choice for the three choices and 6 possibility because you already choose one for the first choice for the second choice of the three choices and 5 possibilities for the third choice of the three choices because you already chose two things in the other two choices. You have to divide by 3 factorial (3!) because mountain climbing, canoeing and video games is the same choice as canoeing mountain climbing and video games and 7x6x5 accounts for all variances in combination even though order doesn't matter in this problem. It is the same for all of the number of choices. Just divide by factorials (number!)

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• Answerer 2 

  57 different ways

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• Answerer 3

  Out of 7 activities listed, the student may check off each activity in either of 2 ways: "Like" or "Dislike."
  
  This makes 2 possibilities for each of 7 activities, for a total of:
  
  2^7 = 128
  
  ... ways in all.

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