There are five orders of ice cream. Each ice cream order has a distinct cone, distinct ice cream flavor, and distinct topping. I have to put on a cone before I can put on ice cream, and I have to put on ice cream before I can put on the topping. I can arrange the orders multiple ways. For example, I can put out all three cones, then put on the ice cream then the toppings. Or I can just build each individual order up one by one. How many sequences can I fulfill the 5 orders? (I can only do 1 action at a time)
So my thoughts are: I have 5 options in the beginning (since there are 5 distinct cones) (So 5 choose 1), then when I choose a cone, I can put on that specific cone's ice cream, or put up another cone. So there are another 5 options. But I'm stuck on what to do from here.
Best Answer
So we can think your question like this: we can choose a incomplete order, then build the next item to be put on the order, then choose a incomplete order, then build the next item to be put on the order, etc.
So as there are $5$ orders, we can number them $1, 2, 3, 4, 5$. Then we have $3$ $1$'s, $3$ $2$'s $3$ $3$'s $3$ $4$'s and $3$ $5$'s. Then we have $\frac{(3\times5)!}{{3!}^5}=168168000$ ways. That is a lot!