Question

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 1

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 $^8C_3$ ways.

i.e. $^8C_3=\frac{8!}{3!(8-3)!}$

$^8C_3=\frac{8\times 7\times 6\times 5!}{3\times 2\times 1\times 5!}$

$^8C_3=\frac{8\times 7\times 6}{3\times 2\times 1}$

$^8C_3=8\times 7$

$^8C_3=56$

The remaining 5 elements will automatically fall into the second partition.

Therefore, the total number of partitions is 56.