Mannequins

A shop has a display of 9 mannequins. 6 dressed in red, three in blue. Assuming all the mannequins are otherwise indistinguishable and that the blue mannequins are not to be stood next to each other, how many arrangements are possible?

mannequins image
Solution

The most elegant solution to this question is only a single line long!

The six red mannequins can be spaced out as _R_R_R_R_R_R_. Each of the underscores represent a possible location for the blue mannequins. The are 7 such positions, therefore, the total number of valid solution is $^7\!C_3 = 35$

Other methods exist and shall be added in the fullness of time!