6,425 views
1 votes
1 votes
What is the number of partition of X={a,b,c,d,e,f}.where 'a' and 'c' are always in same block?

3 Answers

4 votes
4 votes

Here, x={a,b,c,d,e,f} mean cardinality of set x= 6

when a and c occurs together means consider it as a single element.

so now the cardinality of the set becomes 5.

total number of partitions for a set having 5 elements = B5(B= bell number)

and B5=52.

https://en.wikipedia.org/wiki/Bell_number

 

1 votes
1 votes

In case you don't want to leave it as an application of some 'Bell Number', here's a more intuitive solution.

We can partition 5 symbols, ie {A, C}, B, D, E, F in 5 ways that is,

All of them in 1 partition

All of them divided in 2 partitions

Similarly 3, 4, 5 partitions

 

1 and 5 partitions:

 Either have all of them in one partition or each separately. Hence total = 2

 

2 partitions:

Can be divided as 2 elements in one partition and remaining 3 in the other OR 1 element in one partition and remaining 4 in the other. Ways of doings it = $\frac{5!}{3!2!} + \frac{5!}{1!4!} = 10+ 5= 15$

 

3 partitions:

Partitions can be formed in 2 ways. Either {1, 1, 3} or {1, 2, 2} . So ways of doing it = $\frac{5!}{3!2!} + \frac{5!}{2!2!2!} = 10+ 15= 25$

 

4 partitions:

Only one possible way to divide which is {1, 1, 1, 2}. Hence $\frac{5!}{1!1!1!2!3!} = 10$

 

Adding all of them =>

$2+15+25+10=52$

edited by

Related questions

0 votes
0 votes
1 answer
1
1 votes
1 votes
1 answer
2
Hrithik Vashishtha asked Jan 24, 2023
360 views
p ->(q->r). Could you please tell me how it is a tautology?
1 votes
1 votes
1 answer
3
0 votes
0 votes
1 answer
4
Sagar475 asked Jan 15, 2022
283 views
If 2,-4 are the eigen value of a non-singular matrix A and IAI=-8, then the eigen vaule of Adj A are x and -y then the value x+y is ?