The set S = {1, 2, 3, ldots, 12} is to be par | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The set $S$ contains $12$ elements.We need to partition $S$ into three disjoint sets $A, B, C$ of equal size.Since the total number of elements is

Vedclass · Accepted Answer

$\frac{12!}{3!(4!)^3}$

Vedclass · Answer

(C) The set $S$ contains $12$ elements.We need to partition $S$ into three disjoint sets $A, B, C$ of equal size.Since the total number of elements is $12$,each set must contain $12 / 3 = 4$ elements.The number of ways to choose $4$ elements for set $A$ from $12$ is $\binom{12}{4}$.The number of ways to choose $4$ elements for set $B$ from the remaining $8$ is $\binom{8}{4}$.The number of ways to choose $4$ elements for set $C$ from the remaining $4$ is $\binom{4}{4}$.The total number of ways to distribute the elements into three labeled sets $A, B, C$ is $\binom{12}{4} 	imes \binom{8}{4} 	imes \binom{4}{4} = \frac{12!}{4!8!} 	imes \frac{8!}{4!4!} 	imes \frac{4!}{4!0!} = \frac{12!}{(4!)^3}$.Since the sets $A, B, C$ are not labeled (the partition is unordered),we must divide by $3!$ to account for the permutations of the three sets.Therefore,the total number of ways to partition $S$ is $\frac{12!}{3!(4!)^3}$.

The set $S = \{1, 2, 3, \ldots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is

Similar Questions

$A$ bag contains balls of two colours,$3$ black and $3$ white. What is the minimum number of balls which must be drawn from the bag,without looking,so that among these there are two of the same colour?

There are $6$ Men and $8$ Women in a club in which a committee of $5$ people has to be made. What will be the number of ways to select $5$ people if at most one man is selected?

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

There are $6$ men and $8$ women in a club,and a committee of $5$ people must be formed. What will be the number of ways to select $5$ people if only men are selected?

Total number of $6$-digit numbers in which only and all the five digits $1, 3, 5, 7,$ and $9$ appear,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module