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5.8 Multinomial Coefficients and Partitions

5.8 Multinomial Coefficients and Partitions. Ordered Partition Number of Ordered Partitions Unordered Partition Number of Unordered Partitions. Ordered Partition.

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5.8 Multinomial Coefficients and Partitions

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  1. 5.8 Multinomial Coefficients and Partitions • Ordered Partition • Number of Ordered Partitions • Unordered Partition • Number of Unordered Partitions

  2. Ordered Partition • Let S be a set of n elements. An ordered partition of S of type (n1,n2,…,nm) is a decomposition of S into m subsets (given in a specific order) S1, S2,…,Sm, where no two of these intersect and where n(S1) = n1, n(S2) = n2, … , n(Sm) = nm and n1 + n2 +… + nm = n.

  3. Example Ordered Partition • List all ordered partitions of S = {a, b, c, d} of type (1,1,2). • ({a},{b},{c,d}) ({a},{c},{b,d}) ({a},{d},{b,c}) ({b},{a},{c,d}) ({b},{c},{a,d}) ({b},{d},{a,c}) ({c},{a},{b,d}) ({c},{b},{a,d}) ({c},{d},{a,b}) ({d},{a},{b,c}) ({d},{b},{a,c}) ({d},{c},{a,b})

  4. Number of Ordered Partitions • The number of ordered partition of type (n1,n2,…,nm) for a set with n elements is The above is called the multinomial coefficient.

  5. Example Ordered Partition • A work crew consists of 12 construction workers, all having the same skills. A construction job requires 4 welders, 3 concrete workers, 3 heavy equipment operators, and 2 bricklayers. In how many ways can the 12 workers be assigned to the required tasks?

  6. Example Ordered Partition (2) • Each assignment of jobs corresponds to an ordered partition of the type (4,3,3,2). • The number of ordered partitions is

  7. Unordered Partition • Let S be a set of n elements. An unordered partition of S of type (n1,n2,…,nm) is a decomposition of S into m subsets S1, S2,…,Sm, where no two of these intersect and where n(S1) = n1, n(S2) = n2, … , n(Sm) = nm and n1 + n2 +… + nm = n.

  8. Example Unordered Partition • List all unordered partitions of S = {a, b, c, d} of type (1,1,2). • ({a},{b},{c,d}) ({a},{c},{b,d}) ({a},{d},{b,c}) ({b},{c},{a,d}) ({b},{d},{a,c}) ({c},{d},{a,b})

  9. Number of Unordered Partitions • The number of unordered partitions of type (n1,n1,…,n1) for a set with n elements is

  10. Example Unordered Partition • A work crew consists of 12 construction workers, all having the same skills. In how many ways can the workers be divided into 4 groups of 3?

  11. Summary Section 5.8 - Part 1 • Let S be a set of n elements, and suppose that n = n1 + n2 +… + nm where each number in the sum is a positive integer. Then the number of ordered partitions of S into subsets of sizes n1, n2, …, nmis

  12. Summary Section 5.8 - Part 2 • Let S be a set of n elements, where n = m r. Then the number of unordered partitions of S into m subsets of size r is

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