To solve problem number 31, let's go through the details:

Problem Analysis

We have:

• 9 people needing to travel using 2 cars.
• Each car has a capacity of 6 people.
• The condition given is that each car's owner must drive their own car.

Solution Approach

1. Assign the Drivers: Since each car must be driven by its owner, we have 2 people assigned as drivers from the start. This leaves us with 7 people to be distributed between the two cars.

2. Distribute the Remaining 7 People: With each car having a capacity of 6, and each car already having a driver, we now need to assign the remaining 7 people into two groups such that one group joins the driver in one car, and the other group joins the driver in the second car.

3. Choosing People for Each Car:

   • Let’s denote the number of additional people in the first car as $x$, then the other car will have $7 - x$ people.
   • To ensure the total capacity constraint, both $x + 1 \leq 6$ and $(7 - x) + 1 \leq 6$ must hold true (each group along with its driver cannot exceed 6).

   This means that $x$ can be either 3 or 4:

   • If $x = 3$, then the first car will have 4 people (driver + 3 others), and the second car will have 5 people (driver + 4 others).
   • If $x = 4$, then the first car will have 5 people (driver + 4 others), and the second car will have 4 people (driver + 3 others).

4. Calculating the Combinations:

   • If the first car takes 4 people (including its driver), we need to choose 3 out of the remaining 7 people to go in the first car. This can be done in $\binom{7}{3}$ ways.
   • If the first car takes 5 people (including its driver), we need to choose 4 out of the remaining 7 people to go in the first car. This can be done in $\binom{7}{4}$ ways.

5. Calculate Each Combination:

   • $\binom{7}{3} = 35$
   • $\binom{7}{4} = 35$

6. Total Ways:

   • Adding these, we get $35 + 35 = 70$.

Answer

Thus, the number of ways to arrange the 9 people into the two cars, given the conditions, is 70.

Final Answer

The correct answer is: B. 70

Would you like more details or have any questions?

Here are five additional questions related to this problem:

1. How do binomial coefficients work, and why were they used here?
2. What if there were 3 cars and similar conditions; how would the approach change?
3. How can we generalize this solution for any number of cars and people?
4. What other methods could be used to solve combinatorial problems like this?
5. Why is it essential to consider each car's capacity in combinatorial arrangements?

Tip: Always double-check conditions (like capacity limits) in combinatorial problems to avoid overlooking constraints that affect your solution.