Combination or Permutation

When you pick 4 out of 10 people, the order does not matter (whether Bob, Allen, Dick, Ethan go is the same as Ethan, Allen, Dick, or Bob so it does not matter if Bob is position 1 or position 4) so you need to use the formula for combinations, which is (n!) / (k! * (n-k)!) which is (10!) / (6!) * (4!). Once you simplify, the answer is 210. (We assume they're all picked together, versus being picked one by one so that the pool of choices decreases after each pick, which would be a permutation formula and order would matter).

For the second part of the question, the order does matter since each location is different - position 1, 2, 3, 4 - north, south, east, west. Bob in the north corner (position 1) is different from him being in the east corner (position 3). So here, we need to use a permutation formula, which is (n!) / (n-k)! so n=4 positions or choices and we have 4 people to arrange, k=4, and when you simplify, you get 4! which is 24.

The first question is asking the ways to choose 4 people from 10 , so order of arrangement is not taken into consideration , 10C4 = 210 . The second question is asking the ways to separate 4 people into 4 different places , so the order of arrangement has to be taken into consideration . 4P4 = 24.

The first question is asking how many ways are there to choose 4 people from 10. This is precisely $\binom{10}{4} = 210$ The second question is asking for the number of ways to order 4 people in 4 different places; this is precisely $4! = 24$