+295 Four boxes with ball capacity of 3 , 5 , 7 and 8 are given . In how many ways can 19 same balls be put together in these boxes .
Here is my attempt at distributing the 19 balls. Please be warned that this may be wrong! Take it at your own peril !!!!:
1- 8 + 7 + 3 + 1 = 19
2- 8 + 7 + 2 + 2 = 19
3- 8 + 6 + 4 + 1 = 19
4- 8 + 6 + 3 + 2 = 19
5- 8 + 5 + 5 + 1 = 19
6- 8 + 5 + 4 + 2 = 19
7- 8 + 5 + 3 + 3 = 19
8- 8 + 4 + 4 + 3 = 19
9- 7 + 7 + 4 + 1 = 19
10- 7 + 7 + 3 + 2 = 19
11- 7 + 6 + 5 + 1 = 19
12- 7 + 6 + 4 + 2 = 19
13- 7 + 6 + 3 + 3 = 19
14- 7 + 5 + 5 + 2 = 19
15- 7 + 5 + 4 + 3 = 19
16- 6 + 6 + 5 + 2 = 19
17- 6 + 6 + 4 + 3 = 19
18- 6 + 5 + 5 + 3 = 19
The capacity of the boxes is counted from left to right as: 8, 7, 5, 3
1 - above can have one additional arrangement: 8 + 7 + 1 + 3........19
4- above .............................................................: 8 + 6 + 2 + 3........20
10 - above............................................................: 7 + 7 + 2 + 3.........21
11- above..............................................................: 6+ 7 + 5 + 1.........22
12 - above.............................................................: 6 + 7 + 4 + 2........23
13-above................................................................: 6 + 7 + 3 + 3.......24
14- above................................................................: 5 + 7 + 5 + 2.......25
15- above.................................................................:5 + 7 + 4 + 3.......26
16-above..................................................................: 6 + 7 + 5 + 1...... 27
6 + 7 + 4 + 2.......28
6 + 7 + 3 + 3.......29
18-above...................................................................: 5 + 6 + 5 + 3.......30
So, it looks that in addition to the 18 listed above, you can have another 12 arrangements for a grand total =30
+118608 A question to the 2 answering guests.
Why can't one of the boxes be empty?
0,4,7,8
0,5,6,8
0,5,7,7
+26367 Four boxes with ball capacity of 3 , 5 , 7 and 8 are given .
In how many ways can 19 same balls be put together in these boxes .
\(\begin{array}{|r|cccc|} \hline &\text{Box with} &\text{Box with}& \text{Box with}& \text{Box with} \\ &\text{capacity of 8} & \text{capacity of 7}& \text{capacity of 5} & \text{capacity of 3} \\ \hline 1. & 4 & 7 & 5& 3 \\ 2. & 5 & 6 & 5& 3 \\ 3. & 5 & 7 & 4& 3 \\ 4. & 5 & 7 & 5& 2 \\ 5. & 6 & 5 & 5& 3 \\ 6. & 6 & 6 & 4& 3 \\ 7. & 6 & 6 & 5& 2 \\ 8. & 6 & 7 & 3& 3 \\ 9. & 6 & 7 & 4& 2 \\ 10. & 6& 7 & 5& 1 \\ 11. & 7& 4 & 5& 3 \\ 12. & 7& 5 & 4& 3 \\ 13. & 7& 5 & 5& 2 \\ 14. & 7& 6 & 3& 3 \\ 15. & 7& 6 & 4& 2 \\ 16. & 7& 6 & 5& 1 \\ 17. & 7& 7 & 2& 3 \\ 18. & 7& 7 & 3& 2 \\ 19. & 7& 7 & 4& 1 \\ 20. & 7& 7 & 5& 0 \\ 21. & 8& 3 & 5& 3 \\ 22. & 8& 4 & 4& 3 \\ 23. & 8& 4 & 5& 2 \\ 24. & 8& 5 & 3& 3 \\ 25. & 8& 5 & 4& 2 \\ 26. & 8& 5 & 5& 1 \\ 27. & 8& 6 & 2& 3 \\ 28. & 8& 6 & 3& 2 \\ 29. & 8& 6 & 4& 1 \\ 30. & 8& 6 & 5& 0 \\ 31. & 8& 7 & 1& 3 \\ 32. & 8& 7 & 2& 2 \\ 33. & 8& 7 & 3& 1 \\ 34. & 8& 7 & 4& 0 \\ \hline \end{array} \)
