Formation of Groups

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JEE Main 2007

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

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

(B)

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

(C)

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

(D)

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

JEE Main 2021 (24 February Shift 2)

LEVELJEE Main

The students $S_{1}, S_{2}, \dots, S_{10}$ are to be divided into 3 groups $A, B$ and $C$ such that each group has at least one student and the group $C$ has at most 3 students. Then the total number of possibilities of forming such groups is

JEE Main 2023 (10 April Shift 2)

LEVELJEE Main

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

JEE Main 2024 (05 Apr Shift 2)

LEVELJEE Main

Let the set $S = {2, 4, 8, 16, \dots, 512}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = ϕ$ . The maximum number of such possible partitions of $S$ is equal to:

JEE Main 2024 (01 Feb Shift 1)

LEVELJEE Main

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to:

JEE Main 2024 (27 Jan Shift 2)

LEVELJEE Main

Let $α = \frac{( 4 !)!}{( 4 ! ) ^{3!}}$ and $β = \frac{( 5 !)!}{( 5 ! ) ^{4!}}$ . Then :

(A)

α \in N

and

β \in / N

(B)

α \in / N

and

β \in N

(C)

α \in N

and

β \in N

(D)

α \in / N

and

β \in / N

JEE Advanced 2020

LEVELJEE Main

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ____.

JEE Advanced 2024

LEVELJEE Main

A group of 9 students $s_{1}, s_{2}, \dots, s_{9}$ is to be divided to form three teams $X, Y$ and $Z$ of sizes 2, 3 and 4 respectively. Suppose that $s_{1}$ cannot be selected for the team $X$ , and $s_{2}$ cannot be selected for team $Y$ . Then the number of ways to form such teams, is ________.

JEE Advanced 2018

LEVELJEE Advanced

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $α$ is the number of one-one functions from $X$ to $Y$ and $β$ is the number of onto functions from $Y$ to $X$ , then the value of $\frac{1}{5 !} (β - α)$ is ________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set S={1,2,3,…,12} is to be partitioned into three sets A,B,C of equal size. Thus A∪B∪C=S,A∩B=B∩C=A∩C=ϕ. The number of ways to partition S is

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α=(4!)3!(4!)!​ and β=(5!)4!(5!)!​. Then :

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ____.

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

Let $α = \frac{( 4 !)!}{( 4 ! ) ^{3!}}$ and $β = \frac{( 5 !)!}{( 5 ! ) ^{4!}}$ . Then :