The bicycling club rented three vans to take people skiing. Each van could hold
ID: 3141262 • Letter: T
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
Given that 12 vans
each can hold maximum 7 people and each need atleast one
Peter, the leader, said "I dont care who goes i what van, but obviously we need at least one driver in each one."
This means that the van itself is not important, just how people are arranged with respect to each other. the possible ways that people are arranged are:
7 + 4 + 1
7 + 3 + 2
6 + 5 + 1
6 + 4 + 2
6 + 3 + 3
5 + 5 + 2
5 + 4 + 3
4 + 4 + 4
let's chose first case
for a certain set up, say "7+ 4 + 1"
"7" means any 7 people from the 12 are sitting in the first van.
= 12 choose 7
= 792.
In the second van, you can have any 4 of the remaining 5.
= 5 choose 4
= 5.
The last van is always determined by the other only 1 way
As the order of the vans is not important, we can now simply multiply the three values together.
792*5*1 = 3960
There are 3960 ways of arranging the people if you know that: there are 7 in one van, 4 in the second and 1 in the last.
similerly for second case
7 + 3 + 2
total number of ways = 792 * 10*1
=7920
similerly for third case
6 + 5 +1
total number of ways = 924 * 6*1
= 3744
similerly for 4th case
6 + 4 + 2
total number of ways = 924 * 15*1
= 13860
similerly for 5th case case
6 + 3 +3
total number of ways = 924 * 6*1
= 3744 *3 *1
= 11232
similerly for 6th case
5 + 5 +2
total number of ways = 792 * 6*1
= 4752
similerly for 7th case
5 +4 + 3
total number of ways = 792 * 6*1
= 27720
similerly for 8th case
4 + 4 +4
total number of ways = 495 * 70*1
= 34650
Now add the all 8th case to get final result
= 103878 ways
Answer: 103878 ways
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