1. The problem statement, all variables and given/known data 1. A carpenter has 3 identical hammers, 5 different screwdrivers, 2 identical mallets, 2 different saws and a tape-measure. She wishes to hang the tools in a row on a tool rack on the wall. In how many ways can this be done if the first and last positions on the rack are to be mallets and the hammers are not to be next to each other? 2. Relevant equations The number of ways of arranging n objects which include 'a' identical objects of one type, 'b' identical objects of another type,.... is n!/(a!b!....) n objects divided into m groups with each group having G1, G2, ..., Gm objects respectively has m! * G1! * G2! * ... *Gm! 3. The attempt at a solution Since the mallets are identical and there are only 2, we don't have to worry about them. We can reduce the problem to 11 objects to be arranged. Since out of the 11 objects, 3 are identical which are the hammers we have a total of 11!/3! ways of permuting the 11 objects. However we don't want the hammers to be next to each other. So calculate the ways they are next to each other. We have 9 groups of objects. As the hammers are identical and must all be next to each other in a threesome, we have 9! ways of permuting the 11 objects. So we subtract the cases when the hammers are next to each other. 11!/3!-9!=6289920 However the answers suggested 13063680 ways. I can't see what is wrong with my reasoning.