The bicycling club rented three vans to take people skiing. Each van could hold

Question

The bicycling club rented three vans to take people skiing. Each van could hold 7 people. As it turned out, only 12 people could make the trip, but because of the amount of equipment they had to bring, they still needed all the vans. Peter, the leader, said "I dont care who goes i what van, but obviously we need at least one driver in each one." Without regard to which vans the three groups get into or to who is in which group, in how may ways can the 12 people be split up?

For example: 4 people in one van, 4 people in another, and 4 people in another van

Explanation / Answer

Setting up the situation as:

Number of vans: 3

Maximum number of people that, a van can hold = 7

Total number of people = 12

Now arranging possible combination :

Van is not important , we need to know how many ways people can split up.

We set up 8 possible combination with given situation Now priced to find number of ways for each combination .

1.

In first van , choosing 7 people out of 12 , in second van choosing 4 people out of remaining 5 . we can ignore third because it is always determined by other two.

12C7 × 5C4 = 792× 5 = 3960

2.

12C7× 5C3= 792× 10= 7920

3.

12C6× 6C5= 924×6 = 5544

4.

12C6× 6C4= 924× 15 = 13860

5.

12C6× 6C3= 924×20= 18480

6.

12C5× 7C5= 792× 21 = 16632

7.

12C5× 7C4= 792× 35= 27720

8.

12C4× 8C4= 495× 70= 34650

Total number of ways =

3960+7920+5544+13860+18480+16632+27720+34650

= 128766.

Combination Van-1 Van-2 Van-3 1. 7 4 1 2. 7 3 2 3 6 5 1 4. 6 4 2 5. 6 3 3 6. 5 5 2 7. 5 4 3 8. 4 4 4

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