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in how many ways can we distribute 6 pieces of identical candy to 5 kids, if the two youngest kids are twins and insist on receiving an equal number of pieces?

 Dec 31, 2019
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in how many ways can we distribute 6 pieces of identical candy to 5 kids, if the two youngest kids are twins and insist on receiving an equal number of pieces?

 

1:

The twins get 3 pecies each and the others get none.        1 Way

 

 2:

The twins get 2 pieces each and one of the others gets 2 pieces                3 ways

The twins get 2 pieces each and 2 of the others get 1 piece                        3 ways

 

3:

The twins get 1 piece each and one of the others gets 4 pieces               3 ways

The twins get 1 piece each and the others get 3,1, and 0 pieces              3!=6 ways

The twins get 1 piece each and the others get 2,2, and 0 pieces              3 ways

The twins get 1 piece each and the others get 2,1, and 1 pieces              3 ways

 

4.

The twins get 0 and one of the others gets 6 pieces               3 ways

The twins get 0 and the others get 5,1, 0                                6 ways

The twins get 0 and the others get 4,2, 0                                6 ways

The twins get 0 and the others get 4,1, 1                               3 ways

The twins get 0 and the others get 3,3, 0                                3 ways

The twins get 0 and the others get 3,2, 1                                6 ways

The twins get 0 and the others get 2,2, 2                                1 ways

 

So that is  1+3+3+3+6+3+3+3+6+6+3+3+6+1 = 50

 

I could easily have made a careless mistake but my logic is sound.

 Dec 31, 2019

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