How do I solve a probability problem involving permutations and having two steps?

The problem:

A man spends 7 nights in a city. He has a list of the 8 best Italian restaurants and the 9 best Chinese restaurants. How many ways can he eat 7 meals at these restaurants, assuming a different restaurant each night and he wishes to alternate between Italian and Chinese food.

I first tried using permutations using $n=17$ and $r=7$. The result: $98.017.920$. Then knowing I had to do something with alternating the restaurants and assuming that he starts with an Italian restaurant, there would be 4 Italian and 3 Chinese restaurants. So I used permutations for Ital. $n=8,r=4$ and Chin. $n=9,r=3$ for $1680$ and $504$ respectively. I divided the product but I know that is not the answer. I don't have a good grasp of when to use permutations or combinations.

Any help in clearing this up would be greatly appreciated.

Thank you

If the man starts the first night eating in a Chinese restaurant, he has 9 choices. In the second night, for an Italian restaurant, he has 8 choices. In the 3rd night, he has 8 Chinese restaurants left, to eat at. And so on.

This gives: $9\cdot 8\cdot 8\cdot 7\cdot 7\cdot 6\cdot 6$

If he starts eating in an Italian restaurant, he has

$8\cdot 9\cdot 7\cdot 8\cdot 6\cdot 7\cdot 5$

We have to add these to know how many possible ways there are to spend his 7 days eating from different restaurants each night.

• Is there some ambiguity here because some combinations result in eating at the same set of restaurants but in a different order? If he eats at Italian_1 on night 1 and Italian_2 on night 3 isn't that the same as if he ate at Italian_2 on night 1 and Italian_1 on night 3
– JimM
Commented Aug 6, 2018 at 12:37

I suggest another solution:

• Assuming he starts with an Italian restaurant, thus having four Italian and three Chinese meals: $\binom84\cdot\binom93=5880$ possibilities.

• Assuming he starts with a Chinese restaurant, it's $\binom94\cdot\binom83=7056$ possibilities.

This yields a total of 12936 possibilities.

My reasoning is that the days (or nights) do not matter: As suggested by @JimM in his comment, Italian_1 on the first night and Italian_2 on the second is the same result as Italian_2 on the first and Italian_1 on the second -- in both ways, he eats at those two restaurants.

• The first solution is the correct one. The answer was in the back of the book I am working on,but I couldn't get it. I believe that permutations are used since the question asked, " in how many ways can he eat at the 7 restaurants?". Even though (Italian 1 and Italian 2) and (Italian 2 and Italian 1) are the same 2 restaurants, they are another way to order the restaurants. Commented Aug 6, 2018 at 15:31