(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the number of ways n different games can be divided between n different children so that every time, exactly one child gets no game.

2. Relevant equations

3. The attempt at a solution

Here, since exactly one child gets no games, n games are distributed among (n-1) children which gives one child 2 games. Now, since a child can get 2 games out of n in [tex]^n C_2 [/tex] and the children can be arranged in n! ways, to total ways to distribute the games is [tex]^n C_2 * n![/tex]

I was wondering if the following solution is also correct:

[tex]x_1 + x_2 +x_3 +... x_n =n[/tex] such that exactly one [tex]x_i =0[/tex] and exactly one [tex] x_j =2[/tex] where i and j are not the same elements and all other elements are equal to 1.

Therefore, the solution should be:

Coeff. of [tex]x^n[/tex] in [tex] ( (x^0)(x^1+x^2)(x^(n-2) )^n [/tex]

Thats coeff. of [tex]x^n[/tex] in ( (x^0)(x^1+x^2)(x^(n-2) )^n

Is this correct?

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# Find the number of ways n different games can be divided

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