Question

Total number of different partitions of a set having four elements is

Answer 1

So the sum of these 1+7+6+1=15 is the number of total possible partitions of a 4 element set.

hope it is helpfull to you

Answer 2

The second kind of Stirling number, S(n,k), may be used to determine how many different methods there are to divide a given number of n items into k parts. S(4,1) = 1, S(4,2) = 7, S(4,3) = 6, and S(4,4) = 1 are the number of ways to have 4 items in 1 partition, 2 partitions, 3 partitions, and 4 partitions, respectively. The total number of alternative divisions of a set of four elements is therefore given by these: 1+7+6+1=15.

This is also the fifth Bell number (the first being the number of partitions for 0 elements, making the fifth the number of elements), however, one technique to get the Bell number is to sum up all the Stirling numbers of the second sort.

Hence the Total number of different partitions of a set having four elements is 15

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