Let S = {1, 2, ...,8). How many partitions of S are there consisting of exactly two blocks, where one of the blocks has 3 elements and the other block has 5 elements? Answer. 112 Hint:

Answer:The total number of partitions is 56.Step-by-step explanation:Given : Let S = {1, 2, ...,8).To find : How many partitions of S are there consisting of exactly two blocks, where one of the blocks has 3 elements and the other block has 5 elements?Solution : Set S = {1, 2, ...,8)According to question,The first partition consists of 3 elements. Which means 3 elements can be chosen out of 8 in [tex]^8C_3[/tex] ways. i.e. [tex]^8C_3=\frac{8!}{3!(8-3)!}[/tex][tex]^8C_3=\frac{8\times 7\times 6\times 5!}{3\times 2\times 1\times 5!}[/tex][tex]^8C_3=\frac{8\times 7\times 6}{3\times 2\times 1}[/tex][tex]^8C_3=8\times 7[/tex][tex]^8C_3=56[/tex] The remaining 5 elements will automatically fall into the second partition. Therefore, the total number of partitions is 56.