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# HELP

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An old billionaire is writing his will, and deciding how to distribute his nine estates among his three children. He wants to give each child at least one estate, and he does not want to give any two of his children the same number of estates. In how many possible ways can he distribute his estates?

Apr 19, 2020

#1
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First of all, since he wants to give all of his children at least one, you can write it as how many ways to distribute 6 estates among 3 children with no children having the same amount, and this is kind of a small number so you could just bash it honestly.

1 + 2 + 3

0 + 1 + 5

0 + 2 + 4

Are the only choices.

Now there are two different problems here, and you didnt specify, or i didnt catch which one is being asked by the problem. The first is to treat each one as a different child, and see how many ways you can give each child a different estate, idrk how to explain, just like maybe look at the second way, and this one might make more sense. Basically treating the children as different people.

Remember each one can be distributed in 6 ways, because there are three children (3!), so 3 possibilities * 6 ways to distribute the estates, or 18 ways.

The second way is just to count how many different ways you could allocate the estates, in which case its just 3.

Apr 19, 2020
#2
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Sorry for not clarifying, but your answer is incorrect, each of the estates are also counted as distinct.

Guest Apr 20, 2020
#3
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ok, here is my logic,

No promises that it is correct.

Give one property to each child.  9*8*7 = 504 ways.

Now there are 6 properties left to be divided between the 3 children

6,0,0  = 3 ways

5,1,0 = 3!*6 = 36ways

4,1,1 = 3!*6*5/2 = 90ways

4,2,0 =  3!*6C4 = 90ways

3,3,0 = 3*6C3 = 60ways

3,2,1 = 3!*6C3*3 = 360ways

2,2,2 = 6C2*4C2 = 90ways

3+36+90+90+60+360+90 = 729ways

9*8*7*729 = 367,416 ways

I am certainly not going to guarantee that this is correct.

If you get the correct answer guest, then please share it with us. Preferably with the working.

Apr 20, 2020
edited by Melody  Apr 21, 2020
#4
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Here is my attempt:

Number of partitions of 9 into 3 parts.
7 + 1 + 1 = 9 - Not Allowed
6 + 2 + 1 = 9 - Allowed =3! =6 permutations
5 + 3 + 1 = 9 - Allowed =3! =6 permutations
5 + 2 + 2 = 9 - Not Allowed
4 + 4 + 1 = 9 - Not Allowed
4 + 3 + 2 = 9 - Allowed =3! =6 permutations
3 + 3 + 3 = 9 - Not Allowed
6 + 6 + 6 = 18 ways to distibute the estate.

Apr 21, 2020
#5
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You are assuming that all the estates are the same when the asker has stated that they are not.

Melody  Apr 21, 2020
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If each child is to receive AT LEAST one estate, and the estates are DISTINCT, then the formula suggested is “Stirling Numbers of the 2nd kind x 3!”, or: S(k, n) x n! =S(9, 3) x 3!.

Stirling Numbers are calculated as follows:

k=9; n=3; 1/k! * sumfor(i, 0, k,(-1)^i * (k nCr  i) * (k - i)^n =3,025

=3,025 x 3! =18,150 ways of distributing the estate!

Apr 21, 2020