Placing Objects: Englishman, French, Turks In how many ways can you place 6 English, 7 French and 10 Turkish men in a line so that each Englishman is between a French me and a Turkish, and no French men is next to a Turkish man.

A partial answer only.

Start with the six Englishmen. Then we need to fill the gaps between each two consecutive Englishmen, as well as the spaces at both ends of the line. That's seven spaces in all.

In each space, there must be only Frenchmen or only Turks. Furthermore, according to the conditions, they must alternate. Thus the possible patterns are FETEFETEFETEF and TEFETEFETEFET, where each E represents an Englishman, each F one or more Frenchmen, and each T one or more Turks.

For each of these two patterns, you must count the number of ways to distribute the ten Turks and the seven Frenchmen among the spaces allotted to them.