Order of Actions!
Andy exercises every morning for 30 minutes, allocating 10 minutes for each of the 3 exercises he chooses. But these 3 exercises chosen don't have to be distinct, and there are 5 different exercises A, B, C, D, and E to choose from. For example, (E, A, C), (D, D, D), (C, C, A), and (B, A, B) are possible combinations of 3 exercises for a morning. Since order doesn't matter, (E, A, C) for example is considered to be the same combination as (A, C, E).
How many different combinations are there for his 30-minute morning exercise?
PS: This is my first problem so it may be phrased in a confusing way or just plain stupid, so any feedback would be really great.
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The Solution is 35, This is a bit long but hopefully it will be helpful so lets go through this together !
A neat trick which sometimes proves useful is to try and write the solutions and imagine what they might look like and can you write the solution in different ways, we can write them in many ways one way and the simplest would be to just write the exercises he did legs,legs,arms arms,back,legs and so on but you can represent the solution in another way which might lead to some insight and indeed it does,You can write each arrangement like a command composed of either doing the exercise or passing the exercise like ordering a robot,so for example training the arms twice and the legs once would be written like this
Arms, Legs, Upperbody, Lowerbody, Back // just for reference !
✔✔⟶✔⟶⟶⟶
You will see that there is a total of 7 actions preformed, whether he does the exercise or goes to the next exercise, You will see that the sum of all the different correct arrangements of these actions will equal to the number of different routines he could pick and thus equal to the number of days it would take him to do it.
Notice that from all the 7 actions we must have 3 of them to be check marks, so we will chose 3 from 7 to be checkmarks so for example we could chose action #1 or #4 or #5 to be checkmarks, now how many different 3 actions can we chose to be checkmarks, well we can pick one action from 7 actions that is 7 choices and then we can pick another action from the remaining 6 and then another one from the remaining five, so we would have a total of 7x6x5 = 210 choices!
But notice that some of them are redundant so changing actions #1 #2 #3 into checkmarks is the same as changing actions #3 #2 #1 we will see that this is a combination problem meaning that we will have to divide by the number of redundants so that we can get the unique results, so how many different ways can I rearrange #1 #2 #3 or for each set contianing 3 actions how many ways can you rearrange these actions, well by simple logic like earlier it will be 3! so we will divide by 3! to get rid of the redundant answers so 210/3! = 35, and that is our solution. Notice that you could have noticed that this is a combination problem from the beginning and the answer would be simply 7C3 = 35
This is an example of a problem where we are allowed to repeat our choice but order doesn't matter, another example would be that you have 7 flavors of icecream and that you want to scoop 4 flavors notice that you can chose the same flavor more than once (repetition) and that a chocolate vanilla icecream is the same as a vanilla chocolate icecream (No order).
In such problems the formula to use is (n+r-1)Cr where n is the number of things and r is the number of things you chose in our example n=5 and r=3 which will give 7C3 = 35
I owe a great thanks to the following resource if you have time check it out, its a really great website full of amazing insights. https://betterexplained.com/
this is my first problem so it might be a little bad since I haven't had alot of experience writing problems :)