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A child has $5$ locomotive and $20$ different toy coaches (Train carriage). In how many ways he can make different trains, If only locomotive can also be considered as train.

My approach:

If add first coach there are 5 choice for me and now if add second boogey then I've 6 choice and so on, so my answer will be $5 \cdot 6\cdot7\cdot ...\cdot 24$.

Now I'll add $5$ to above calculated answer as only locomotive alone can be considered as train.

But given answer is only $5\cdot6\cdot7...\cdot24$

Why they have not added $5$?

Is above solution correct?

N. F. Taussig
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mathophile
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  • What are "boogies"? – Paul Sinclair Feb 09 '21 at 00:27
  • Can multiple locomotives be put on the train? After all real trains have them quite often. Assuming the answer is no, I don't see this calculation at all. Why out of 20 boogies do you only have five choices for the first? Why of the 19 remaining boogies would you only have 6 choices for the second? I would calculate $5$ choices for engine, and $\sum_{n=0}^{20}\frac {20!}{n!}$ choices for the added boogies, for $5\cdot 20!\cdot \sum_{n=0}^{20}\frac 1 {n!}\approx 5\cdot 20!\cdot e$ choices overall. – Paul Sinclair Feb 09 '21 at 00:44

1 Answers1

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Let's consider a much simpler version of the problem. Say we have two locomotives and two coaches. Let's name the locomotives $L_1$ and $L_2$ and the coaches $C_1$ and $C_2$.

We have two ways to append coach $C_1$ to a locomotive. Once we have placed coach $C_1$, there are three ways to place coach $C_2$. We can append it to the other locomotive, place it behind coach $C_1$, or place it between the locomotive to which coach $C_1$ is attached and coach $C_1$. This gives $2 \cdot 3 = 6$ possible arrangements. They are:

$(L_1C_1C_2, L_2)$

$(L_1C_1, L_2C_2)$

$(L_1C_2C_1, L_2)$

$(L_1, L_2C_1C_2)$

$(L_1C_2, L_2C_1)$

$(L_1, L_2C_2C_1)$

Notice that these arrangements include the cases in which a locomotive is by itself. There is no need to handle the cases of an isolated locomotive separately since choosing a position for each coach completely determines the arrangement of the cars on each train.

N. F. Taussig
  • 76,571
  • Thanks, Where can i learn more about combinatorics. I can do some basic problem. I need to learn about partition, generating function, and some simple combinatorics problem too. How did you learn all these?. Can you suggest me some book which i should follow in sequence. – mathophile May 29 '22 at 12:28
  • Also combinatorics problem involving $f(f(x))=x$ from set $A{0,1,2,3,4,5,6,7,8}$ to $A{0,1,2,3,4,5,6}$ – mathophile May 29 '22 at 12:29
  • The first book I read was Mathematics of Choice by Ivan Niven, but it is a bit basic for your purposes. It sounds like what you need is an advanced undergraduate text on combinatorics such as Miklos Bona's A Walk Through Combinatorics. See recommendations here, here, and here. – N. F. Taussig May 30 '22 at 14:35