Red and green balls
There are 2016 boxes numbered 1, 2, 3....2015, 2016. Each box is to be filled up either with a green ball or a red ball in such a way that at least one box contains a green ball and the boxes containing green balls are consecutively numbered. Calculate the total number of ways in which this can be done.
The answer is 2033136.
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1 solution
The following are the various cases :
1 box has green ball. Can be in any of the 2016 boxes. This is possible in 2016 ways.
2 boxes have green balls. The boxes may be numbered as 12, 23, 45, 56......, 20142015, 20152016. Totally 2015 ways.
3 boxes have green balls. Similar to the above, totally there are 2014 ways.
....
All the boxes have green balls. Only one way
Therefore, the total number of ways = 2016 + 2015 + 2014 + 2013 + ...... + 3 + 2 + 1 = 2033136