Possible outcomes of rolling dice

[kelvin macks]

for second group, why the set of numbers of ( 132, 152, 153, 154, 213, 214 , 215, 216, 231, ...) is not included? since they are all three different numbers. https://www.flickr.com/photos/123101228@N03/14323629833/

 

[Adithyan]

for second group, why the set of numbers of ( 132, 152, 153, 154, 213, 214 , 215, 216, 231, ...) is not included? since they are all three different numbers. https://www.flickr.com/photos/123101228@N03/14323629833/

The dies are identical.
132 is nothing but 123; 152=125.. etc. Since, 132 and 123 has the same OUTCOME, i.e, there are three different number on the dies and since the dies are identical, it doesn't matter which dies shows which number as long as the numbers are different.

(NOTE: You would've been right if that question asked to calculate the 'probability' instead of no. of 'outcomes')

Hope this helps! :)

-Adithyan

 

[Simon Bridge]

All those combinations are shown. Order is not important.

 

[kelvin macks]

All those combinations are shown. Order is not important.

why order is not important? i can't understand the question doesn't state whether it's permutation or combination. I'm confused.

 

[Adithyan]

why order is not important? i can't understand the question doesn't state whether it's permutation or combination. I'm confused.

The order is not important because the question asks for the different OUTCOMES of the dies. So, getting 123 and 321 will have the same outcome for the event i.e, the dies show different numbers. So the question clearly states combination.

 

[kelvin macks]

The order is not important because the question asks for the different OUTCOMES of the dies. So, getting 123 and 321 will have the same outcome for the event i.e, the dies show different numbers. So the question clearly states combination.

how if the question states number of outcomes? is it question of permutation?

 

[FactChecker]

The example question is not very clearly worded. Perhaps by saying the dice are identical, the author intends to imply that they cannot be assigned an order. In that case it is better to explicitly state that order does not matter. People often say things are "identical" even when order counts.

 

[Simon Bridge]

why order is not important? i can't understand the question doesn't state whether it's permutation or combination. I'm confused.

The author says that the dice are "identical" ... which usually means that you cannot tell them apart. If you, for example, rolled them one at a time, then you can tell them apart: one die is first, another second and so on. So they are not identical. Therefore, if you know the order the numbers appear, the dice are not identical. Conversely, if the dice are identical, then you do not know the order they appear.

The author reinforces this interpretation by also stating an interest in "different outcomes" - the outcome "1,2,3" means "one die shows a 1, and one die shows a 2, and one die shows a 3". The outcomes "3,2,1" is "one die shows a 1, and one die shows a 2, and one die shows a 3" is the same outcome.

If the author intended that you should treat 1,2,3 and different from 3,2,1, then they would have given an extra definition like saying that the outcomes are represented by ordered triples.

Lastly, the author has provided examples which show you what is intended.

IRL: almost all your problems will be worded in regular language, which tends to be a bit vague compared with rigorous maths language. Part of what you are training to do is to be able to translate real-language problems into maths for analysis. To do this you will have to use the clues that are available to you - like the context of the problem - to determine what is important. It's a value judgement and something of an art-form, one that you must learn if you are to get good at this.

Your lesson here is that when someone says that the dice are "identical", then this is what it means.

As you do more of these problems you will get used to the kind of thinking needed.

In that case it is better to explicitly state that order does not matter. People often say things are "identical" even when order counts.

This is true - however, they seldom say that dice are identical when the order counts, without also saying something about the order being important. This is especially the case in maths exercises. Context is everything - learn to use the metadata.

You can still get it wrong of course - this is why it is good practice IRL to list the assumptions and interpretations where several interpretations are possible. This particular wording for a math problem is actually pretty common. Very few people use rigorous speech even in technical situations.