i think i'm missing something here, if arrangement/order is a part of this problem why is it considered a combination problem as opposed to a permutation one?
It's not advisable to tag the questions using its wording. Just because the question uses the word 'arrangement', it doesn't make this an arrangement problem. It is a combination problem and here is why: When you pick the cards, there is only one way in which you can arrange them - the ascending order which will be unique for any 4 cards you pick. Say, I picked up 3, 5, 6, 10. The only way I can arrange them is this: 3, 5, 6, 10. I cannot arrange them as 3, 6, 10, 5 or 6, 5, 3, 10 or any other arrangement that you can have with 4 unique cards. The number of arrangements here is not 4!. Instead it is only 1 since they must be put in ascending order.
So all you have to do is find out the number of ways in which you can pick 4 cards out of 8 unique cards. This will be done in 8C4 = 70 ways.
The book uses the basic counting principle. Put 4 cards in 4 places in 8*7*6*5 ways and since you can arrange them in only one way, divide by the number of arrangements i.e. 4!
When will it be an arrangement problem? "She has students choose 4 cards randomly, then arrange the cards in ANY order."
Now each selection will have 4! distinct arrangements.
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