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How many ways are there to divide a group of 6 friends among the basketball team, the soccer team, and the track team? (Each team could have anywhere from 0 to 6 of the friends on it. Assume the friends are distinguishable.)

 Oct 20, 2022
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How many ways are there to divide a group of 6 friends among the basketball team, the soccer team, and the track team? (Each team could have anywhere from 0 to 6 of the friends on it. Assume the friends are distinguishable.)

Possibilities

0,0,6          1 way to seperate. 3 alternatives for teams  =3

0,1,5         6C1 way to seperate. 3! alternatives for teams  =36

0,2,4         6C2  way to seperate. 3! alternatives for teams  =36

0,3,3         6C3/2  way to seperate. 3! alternatives for teams  =18

1,1,4         6C4  way to seperate. 3! alternatives for teams  =90

1,2,3         6C3*3  way to seperate. 3! alternatives for teams  =360

2,2,2         6C2*4C2/6  way to seperate. 3! alternatives for teams  =90

 

3+36+36+18+90+360+90 = 633

 

I could easily have made a mistake.  If you get the correct answer, please share it here.

 Oct 21, 2022

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