Seven different coloured lights are fixed on a mast to produce signals by illuminating them. How many signals can be produced, if any number of lights is used?

Since the different combinations (both selection and placement) would produce different signals, we need to consider each case.

Case 1: When only 1 light is used

Here, we need to select and arrange 1 light out of seven. It can be done in 7 ways. However, two produce signals, we need at least two lights. Hence, we shall discard this case. 

Case 2: When 2 lights are used

This can be done by selecting two lights out of seven and then arranging them, which is $$7c_2\times\ 2!$$, which is 49 ways

Case 3: When 3 lights are used

This can be done by selecting three lights out of seven and then arranging them, which is $$7c_3\times\ 3!$$, which is 210 ways

Case 4: When 4 lights are used

This can be done by selecting four lights out of seven and then arranging them, which is $$7c_4\times\ 4!$$, which is 840 ways

Case 5: When 5 lights are used

This can be done by selecting five lights out of seven and then arranging them, which is $$7c_5\times\ 5!$$, which is 2520 ways

Case 6: When 6 lights are used

This can be done by selecting 6 lights out of seven and then arranging them, which is $$7c_6\times\ 6!$$, which is 5040 ways

Case 7: When 7 lights are used

This can be done by selecting 7 lights out of seven and then arranging them, which is $$7c_7\times\ 7!$$, which is 5040 ways

Adding Case 2 till Case 7, we get the answer as 13,699