Question

slader How many partitions of the set {1, 2, 3, . . . , 100} are there such that there are exactly three parts and elements 1, 2 and 3 are in different parts?

Answer 1

Answer:

$3^{97}$ partitions.

Step-by-step explanation:

As we know if we have x number of elements we can arrange them in x! ways.

here we have 100 elements so we can arrange them in 100! ways.

Now if we have exactly three parts and 1,2 and 3 are in different parts then let consider 1 goes to part 1, 2 goes to part 2, and 3 goes to part 3. For the remaining 97 elements,  we have three options for each element, thus we can arrange the remaining 97 elements in $3^{97}$ ways/partitions.

each of these partition will contain only three elements.