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Maps $g$ maps $\left\{1,2,3,4,5\right\}$ onto $\left\{11,12,13,14\right\}$ and $g(1)\neq g(2)$. How many g are there.

My answer: I transformed the question to a easy-understand way and find out the solution. Consider there are five children and four seats. Two of them are willing sitting together but only two of them never seat together.

$$\left(\begin{pmatrix} 5 \\ 2 \end{pmatrix}-1\right)*4!=456$$

However the answer is 216. I don't know what's wrong.

Could you please help me find out what's wrong or give a right way to solve the problem?

Thanks!

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2 Answers 2

up vote 3 down vote accepted

Suppose that $g$ is a function from $\{1,2,3,4,5\}$ onto $\{11,12,13,14\}$. Then exactly one of the numbers in the set $\{11,12,13,14\}$ is $g(k)$ for two different values of $k$. In terms of the children and the seats, exactly one seat has to contain two children. There are $4$ ways to choose this seat. There are $\binom52-1=9$ ways to choose which pair of children will occupy the seat. The remaining $3$ children must each take one of the other $3$ seats, and they can do this in $3!$ ways. The final result is therefore $$4\cdot\left(\binom52-1\right)\cdot3!=4\cdot9\cdot6=216\;.$$

In other words, the only thing wrong with your answer is the arithmetic:

$$\left(\binom52-1\right)\cdot4!=216\;,$$

not $456$.

(I like your translation into a problem about children and seats.)

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Thanks a lot : ) –  J.A.F Oct 29 '12 at 16:22
    
@PENGTENG: You’re welcome. –  Brian M. Scott Oct 29 '12 at 16:22
    
Nice solution Brian M. Scott.. –  juantheron Nov 15 '13 at 3:37

EDIT: Oh, ”onto” means it's surjective. In that case Brian M. Scott has the right answer, tough you can still solve it my way.

Old answser: Here's a tip for how I would solve it: How many functions are there from the first set to the second without restrictions? How many are there for $g(1)=g(2)$?

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Have you read my answer? What's your opinion? Thanks : ) –  J.A.F Oct 29 '12 at 16:05
1  
I see how you think now, you miss the functions that leave some chairs empty (child 1,3,4,5 sits in chair 11 and child 2 in chair 12, for example). –  Max Morin Oct 29 '12 at 16:08
    
In my opinion,if there are chairs are empty, it is a contradiction with "onto". –  J.A.F Oct 29 '12 at 16:24
    
@PENGTENG Right you are, I am not used to that terminology. :) –  Max Morin Oct 29 '12 at 16:26

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