10. There are thirty people: ten speak only English, ten speak only Spanish, and

Question

10. There are thirty people: ten speak only English, ten speak only Spanish, and ten speak only French. There are five chairs in row. How many ways can we put five people in the chairs such that no two people sitting next to eachother speak the same language. Count a 'way' as being one arrangement of English, Spanish, and French speakers, so for example (S, E, S, E, Sy counts as one way Hints (a) You get the same answer if you start with a different number of people provided you have at least 5 English, 5 Spanish, and 5 French speakers. (b) It might help you to test whatever method you are thinking about using on a smaller number of chairs. If we let n be the number of chairs, then we have n-5. What if n 3? You can test your method with n-3 because it is easy to write down all of the different ways for 3 chairs and then just count them.

Explanation / Answer

Since we are considering the sequence based on only the language of the person, we can safely assume that all people speaking same language are identical.

Now, for the first chair, we can select either of the 3 languages and can select any one of the persons who speak the selected language.

Then, for the second chair, we have just two choices, as the the selected language cannot be chosen once again, due to the constraint that no two people sitting next to eaach other speak the same language.

For the third chair, the constraint on the language selected for first chair is removed, but the language selected for the second chair is unavailable. So we hav two choices left.

Similarly, there will be two choices for the fourth and the fifth chair.

Thus, the total number of ways in which five people can be put in the chairs with the constraint that no two people sitting next to eaach other speak the same language is 3 * 2 * 2 * 2 * 2 = 48.

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