Advent calendar 2025
18 December
There are 5 different ways to make a set of numbers between 1 and 5 such that the smallest number in the set is equal to the number of numbers in the set. These 5 sets are: {1}, {2, 3}, {2, 4}, {2, 5} and {3, 4, 5}.
How many ways are there to make a set of numbers between 1 and 14 such that the smallest number in the set is equal to the number of numbers in the set?
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The sets of numbers between 1 and \(n\) are either also valid sets with numbers between 1 and \(n-1\); or they can be made by taking a valid set of numbers
between 1 and \(n-2\), adding 1 to each number and appending an \(n\) to the set. Therefore the number of sets is the sum of the previous two terms (ie it's the Fibonacci sequence).
Using this, we can work out that the number of sets of numbers between 1 and 14 is 377.
