4. In how many ways can seven different items (3 weighing 2 tons each, 4 weighing 1 ton each) be loaded into two trucks with capacities of 6 tons and 5 tons, if the arrangement of the items inside the trucks does not matter? (12 points)

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# Solution.

Solution. The load can be distributed as $6+4$ or $5+5$.

| 6 tons | 4 tons | number of ways | 5 tons | 5 tons | number of ways |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $2+1+1+1+1$ | $2+2$ | $C_{3}^{1}=3$ | $2+1+1+1$ | $2+2+1$ | $C_{3}^{1} \cdot C_{4}^{3}=12$ |
| $2+2+1+1$ | $2+1+1$ | $C_{3}^{2} \cdot C_{4}^{2}=18$ | $2+1+2+1$ | $2+1+1+1$ | $C_{3}^{2} \cdot C_{4}^{1}=12$ |
| $2+2+2$ | $1+1+1+1$ | 1 | $2+2+1$ |  |  |

## Answer: 46