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  • The number of ways to partition a set of n objects into r non-empty subsets: Stirling number of the second kind, S(n, r) where n=10 and r=3  Optio

The number of ways to partition a set of n objects into r non-empty subsets: Stirling number of the second kind, S(n, r) where n=10 and r=3

 

Option: 1

62


Option: 2

88


Option: 3

28


Option: 4

56


Answers (1)

best_answer

To find the number of ways to partition a set of 10 objects into 3 non-empty subsets, we can calculate the Stirling number of the second kind, S(10, 3).

Using the formula for the Stirling number of the second kind:

\mathrm{S(n, r)=r \times S(n-1, r)+S(n-1, r-1)}

and the base cases:
\mathrm{ S(n, 1)=1}
\mathrm{ S(n, n)=1}
we can calculate  S(10, 3) as follows:
\mathrm{ S(10,3)=3 \times S(9,3)+S(9,2)}
To find S(10,3), we can use the formula \mathrm{ S(n, r)=r \times S(n-1, r)+S(n-1, r-1)}.

Using this formula, we have:
\mathrm{ S(10,3)=3 \times S(9,3)+S(9,2)}
To calculate S(9, 3), we can use the same formula:
\mathrm{ S(9,3)=3 \times S(8,3)+S(8,2)}

Continuing this process, we can calculate the Stirling numbers until we reach the base case.

Using combinatorial methods, we can calculate the number of ways to partition a set of 10 objects into 3 non-empty subsets directly:

We have 10 objects, and we want to partition them into 3 non-empty subsets.

First, we select the objects for the first subset. We can choose any number of objects from 1 to 8 (since we need at least 1 object for each of the 3 subsets). Let's say we choose k objects for the first subset.

Next, we select the objects for the second subset. We can choose any number of objects from 1 to (10-k-1) (since we need at least 1 object for the second subset and we have already chosen k objects for the first subset).

The remaining objects will go to the third subset.

Therefore, the number of ways to partition a set of 10 objects into 3 non-empty subsets is:

\mathrm{ \sum_8^{k=1 \operatorname{ta} 8} \sum_{10-k-1}^{j=1} 1=\sum_{10-k-1}^{k=1 \operatorname{ta} 8} 1=7+6+5+4+3+2+1=28}
\mathrm{ So, S(10,3)=28}.
 

Posted by

jitender.kumar

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