Counting Methods: There are 16 kids to place into 4 different vans. Van 1 can hold 5 kids, Van 2 can hold 2 kids,...

[yuvalenthungavelu699@gmai]

There are 16 kids to place into 4 different vans. Van 1 can hold 5 kids, Van 2 can hold 2 kids, Van 3 can hold 6 kids and Van 4 can hold 3 kids. How many ways can the 16 kids be placed in the vans?

 

[stapel]

There are 16 kids to place into 4 different vans. Van 1 can hold 5 kids, Van 2 can hold 2 kids, Van 3 can hold 6 kids and Van 4 can hold 3 kids. How many ways can the 16 kids be placed in the vans?

Please re-read the "Read Before Posting" message, and reply with the requested information, starting with your thoughts on how to approach this exercise.

Thank you!

 

[Dr.Peterson]

There are 16 kids to place into 4 different vans. Van 1 can hold 5 kids, Van 2 can hold 2 kids, Van 3 can hold 6 kids and Van 4 can hold 3 kids. How many ways can the 16 kids be placed in the vans?

Start by putting 5 kids in van 1. How many ways can you do that?

Then continue. The more you show of your own thinking, or of what you have learned, the better we can help you solve it yourself.

 

[The Highlander]

There are 16 kids to place into 4 different vans. Van 1 can hold 5 kids, Van 2 can hold 2 kids, Van 3 can hold 6 kids and Van 4 can hold 3 kids. How many ways can the 16 kids be placed in the vans?

We don't just give out "answers" to problems posted in the forum; we need to see some effort from a member in their post that shows us what they have tried before seeking help; that also gives us an indication of what ability level they may already possess and thus what might be the best kind of help to offer (otherwise, we don't really know where to start! ?‍).

That is why you have just been given a (useful) hint by @Dr.Peterson but (I believe) he has assumed something (given the direction in which he appears to have pointed you) so I will add some further advice (based on the assumption that you don't already know very much about how to solve this kind of problem but are quite good with numbers and logic; perhaps at the level of a 14/15 year old or above?)

The first thing to note is that the answer will depend on whether (or not) the order in which the children sit in the vans matters!

Your question does not say anything about this and a definitive answer cannot be given unless this is known!
Therefore, I would expect you may have to work out the answer for both cases. (I suspect @Dr.Peterson has already assumed the latter case, ie: where it does not matter ?).

Do you know anything about Combinations & Permutations? This web page will help you if you don't.
Please study it carefully (or read the rest of this post and then study it before attempting the problem yourself).

Consider now the first case, ie: that it does matter which order they sit in.
A picture may help you to visualize this situation.

Lets start by assuming that Van 1 is a Red Van with 5 Red Seats in it, Van 2 is a Green Van with 2 Green Seats in it, Van 3 is a Blue Van with 6 Blue Seats in it and Van 4 is a Yellow Van with 3 Yellow Seats in it.

You could take all of the seats out of the vans and set them up in a line on the street (or two lines as in my picture, below). You could then number each seat as shown; NB: the colours are not relevant for the numbers, I could just as easily have made all the numbers black (probably should have done, on reflection ?).

Now let us further assume that all the children have different names but the first letter of their names span the first 16 letters in the (English) alphabet (A, B, C, D, ......, N, O, P), so we have (for example): Abdul, Bernice, Charlotte and Deepak,... all the way down to: Nick, Olive & Pablo. (This isn't quite so important but may help you to 'picture' the situation in your mind.)

​

Now, if we are dealing with the situation where the order they sit in does matter (this means that if Abdul is in Seat 1, then Deepak in Seat 2 gives an entirely different outcome from putting Bernice in that seat and so on), then you could put any child on Seat 1 which means there are 16 ways you could fill that first seat but for each of those 16 possibilities there are now 15 children left and any one of those could go into Seat 2 which means there are 16 × 15 (= 240) ways you could already fill just the first two seats! When it comes to filling the third seat there are now 14 children left, any one of which might sit in it giving you now 240 × 14 (= 3,360) ways just to fill the first three seats when the order they sit in does matter.

Do you see the pattern developing here? (The web page will help you understand further).

However, the situation becomes quite different if the order in which the kids sit does not matter. Again, reading through that web page (and @Dr.Peterson's suggestion) may help you to figure out how to tackle that.

Please come back and show us your attempt(s) to arrive an answer (for each case?). Further assistance will then be offered if you don't get the right answers or your working goes off course.

Hope that helps. ?

 

[Steven G]

I add another hint. In how many ways can you arrange the 16 kids. What should you do to this number? Try using the other hints.

 

[Dr.Peterson]

That is why you have just been given a (useful) hint by @Dr.Peterson but (I believe) he has assumed something (given the direction in which he appears to have pointed you)

(I suspect @Dr.Peterson has already assumed the latter case, ie: where it does not matter ?).

Actually, I assumed nothing. I intentionally asked a question to elicit information both about what the OP knows, and how they understand the meaning of "placing kids in a van". My suggestion can be used for either interpretation.

I try not to over-help by giving more than they may need on what is likely the wrong problem. And I tend to assume that problems are assigned to students who are (nominally) prepared to solve them.

On the other hand, I do have an opinion: As I read the problem, since no mention is made of distinct locations in a vehicle (though that is certainly possible), and since we don't usually assign seats in such a situation, I would expect kids and vans to be distinguishable, but seats not. This also makes it a far more interesting problem than the other.

Incidentally, a search, which I sometimes do to see if someone already got an answer, reveals that the identical problem was submitted to us and another site in 2020; the other gave a full answer to both interpretations, while we did just what our first two answers here did this time (and an esteemed "deleted member" appears to have agreed with my preferred interpretation).