Final answer:
To find the number of ways six people can be allocated to three different cars with the man alone, we consider the man's three choices and then partition the five women into the remaining two cars in all possible ways, then multiply by the man's choices.
Step-by-step explanation:
The question asks for the number of different ways six people can be allocated to three different cars, ensuring the man is alone. Since there are three cars, the man has three choices of cars to be alone in. After the man has chosen his car, we have five women and two remaining cars. The five women can be distributed in these cars in a 5-partition of two numbers, which means the number of ways we can divide five into two groups without considering order.
The possible partitions for the five women are (3,2), (4,1), and (5,0). For each partition, there are combinations of woman that can occupy the cars. For example, for partition (3,2), we have 5 choose 3 ways to choose which three women are in the first car and the remaining two automatically go to the second car. We repeat this process for each partition. Finally, we must multiply the number of ways for each partition by the three initial choices the man had to get the total number of combinations.