The problem is 294C.

The tutorial says this:

The third sample is ...#...#... where # is a switched on lamp and . is a switched off lamp. As you can see we have three different types of lights. The first three lights (Type A), the 5th to 8th lights (Type B) and the last three lights (Type C). We have to switch on the lights three times for each type of lights. Aside from the order of moves for each type there are possible permutations of the string AAABBBCCC which tells us how to combine the steps of different types. Switching on the lights is done uniquely for types 1 and 3. But for type 2 each time we have to possible options until we're left with one off light. So there are 2^(3 - 1) ways to do this. So the answer would be 1680*1*4*1 = 6720.

I can not understand the part. A little help would be very much appreciated.