maths entrance bmo

Question

during an early morning drive to work ,Simon encountered 'n' sets of traffic lights each set being red , amber or green as he approached it .He noticed that consecutive lights were never the same colour .

given that he saw at least two red lights find a simplified expression in terms of 'n' for the number of possible sequence of colours simon could have seen

Answer 1

Hi Swathi! We can look at this problem in terms of basic probability. Every stop light has three possibilities for its color: red, amber or green. However, if no two consecutive stoplights are ever the same color, then every stoplight (aside from the very first light) only has two possible color choices. For every stoplight that Simon encounters, the number of total possible sequences is then multiplied by 2.

If Simon has encountered at least two red lights, he must have encountered at least three stoplights. This would mean the first and third lights are red (since the second light can't be red -- no two consecutive lights can be the same color).

Thus, the amount of possible sequences Simon can encounter (given that Simon has encountered three or more stoplights) is 3×(2^(n-1)), n ≥ 3.

The 3 represents the first stoplight that has three possible color options. From then on, every stoplight has two options.

Hope this helps! If you have any questions, feel free to ask.