A group of 12 people is traveling in three vehicles. If 4 people ride in each vehicle, in how many different ways can the people be assigned to the vehicles?
A)11,880
B)495
C)34,650
D)11,550

Question

Quizplus · Accepted Answer

The problem can be solved using the concept of combinations since the order of selection does not matter. We have 12 people and we want to divide them into groups of 4 to be placed in 3 vehicles. First, we select 4 people out of 12 to go into the first vehicle, then 4 out of the remaining 8 for the second vehicle, and the last 4 automatically go into the third vehicle. The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of items, and k is the number of items to choose. The calculation is as follows:For the first vehicle: C(12, 4) = 12! / [4!(12-4)!] = 495 ways.For the second vehicle: C(8, 4) = 8! / [4!(8-4)!] = 70 ways.The last 4 people will automatically go into the third vehicle, so we don't need to calculate that.However, since these selections are happening simultaneously and are independent of each other, we multiply the number of ways to find the total number of combinations: 495 * 70 = 34,650.Therefore, there are 34,650 different ways the 12 people can be assigned to the 3 vehicles.

A Group of 12 People Is Traveling in Three Vehicles