Formation of Groups

Formation of Groups

Edited By Komal Miglani | Updated on Jul 02, 2025 07:53 PM IST

Formations of the group are used to find the number of ways n distinct objects can be divided into m groups, whose sizes are known. The number of ways in which (m + n) distinct objects are divided into two groups of the size m and n is equivalent to the number of ways in which m objects are selected from (m + n) objects. The other group of n objects is formed from the remaining n objects.

This Story also Contains
  1. What is Formation of Groups?
  2. Formation of Groups for (m+n+r) Distinct objects
  3. Solved Example Based on Formation of Groups
Formation of Groups
Formation of Groups

In this article, we will learn about the Formation of Groups. This topic falls under the broader category of Permutations and combinations, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE.

What is Formation of Groups?

Formations of the group are used to find the number of ways n distinct objects can be divided into m groups, whose sizes are known.

Consider that 12 people have to be divided among three groups of unequal sizes such as one group has 3 members, one group has 4 members and one group has 5 members.

We could have formed a group of 3 members in ${ }^{12} \mathrm{C}_3$ ways. Having formed a group of three, we would be left with $12-3=9$ people. A group of 4 members can be formed from these 9 members in ${ }^9 \mathrm{C}_4$ ways. For each group of 3 members formed earlier, there would be further ${ }^9 \mathrm{C}_4$ ways of forming a group of four. Thus, the total possible number of ways of forming a group of 3 and a group of 4 would be ${ }^{12} \mathrm{C}_3 \times{ }^9 \mathrm{C}_4$. Now there would be 5 people left who are the third group i.e. the third group can be formed in only 1 way. To maintain consistency, we will say that the third group can be formed in ${ }^5 \mathrm{C}_5$ ways (which is 1 anyway). Thus, the total number of ways of forming the groups is ${ }^{12} \mathrm{C}_3 \times{ }^9 \mathrm{C}_4 \times{ }^5 \mathrm{C}_5$. On expansion, this equals $\frac{(12)!}{3!4!5!}$.

Formation of Groups for (m+n+r) Distinct objects

  • This concept can be generalized for (m+n+r) distinct objects which have to be grouped into three unequal groups containing m,n, and r objects. So this grouping can be done in $\frac{(\mathrm{m}+\mathrm{n}+\mathrm{r})!}{\mathrm{m}!\mathrm{n}!\mathrm{r}!}$ number of ways
  • This same concept will apply for (m+n) distinct object which has to be grouped into two unequal containing m, and n items. Number of ways of dividing mn objects into m groups such that all groups contain n objects equals $\frac{(\mathrm{mn})!}{(\mathrm{n}!)^{\mathrm{m}}} \times \frac{1}{\mathrm{~m}!}$

Example: How many ways 12 people can be divided into 3 groups, such that all three of them contain 4 people each?

Solution: The number of ways of forming the three groups is ${ }^{12} \mathrm{C}_4 \cdot{ }^8 \mathrm{C}_4 \cdot{ }^4 \mathrm{C}_4$. Since we are multiplying these three factors, we are inadvertently also arranging the groups in a particular order. (Remember that if one position can be filled in 5 ways, another can be filled in 4 ways, and the third can be filled in 3 ways when we apply the rule of AND i.e. $5 \times 4 \times 3$, we are basically finding the number of "arrangements" of the three positions)

But the question requires us to just form groups and we do not have to “arrange” the groups. Since we arranged 3 objects that did not have to be arranged, we counted each unique way of forming the groups 3! times i.e. 6 times. Thus, the correct answer would be found by dividing the earlier found answer by 3! This will give you the above formula itself.

Recommended Video Based on Formation of Groups:


Solved Example Based on Formation of Groups

Example 1: The number of ways in which 10 identical balls can be distributed among 5 students so that each gets at least one is

Solution: Rule for distribution of identical objects into Groups:

The number of ways in which $n$ identical things can be distributed to $r$ people (when each gets at least one) is ${ }^{n-1} C_{r-1}$

Now,
The number of ways $={ }^{10-1} C_{5-1}={ }^9 C_4$.
Hence, the answer is ${ }^9 C_4$

Example 2: Out of a group of 8 students, 5 are to be arranged in a queue, among which particular 3 out of 8 must be there. Then the number of ways in which a queue can be formed is

Solution: According to the question, out of 8 students, 3 students are already fixed.
So out of the remaining 5 - and two more have to be included
No. of ways of selecting those two $={ }^5 C_2=10$
Now we have a total of 5 students, so these five can be arranged in 5! i.e 120 ways
$\therefore$ Total number of ways $=120 \times 10=1200$

Now we have a total of 5 students, so these five can be arranged in 5 ! i.e 120 ways
$\therefore$ Total number of ways $=120 \times 10=1200$
Hence, the answer is 1200
Example 3: The number of ways of allotting three rooms in a hotel among 12 guests, where one room has a capacity of 3 , the second has a capacity of 4 , and the third has a capacity of 5 is

Solution: Now,

Firstly, 12 people have to be divided into three groups of 3,4 and 5
No. of ways of grouping $=\frac{12!}{3!4!5!}$
After division allotment can be done only in one way

$
\text { Total number of ways }=\frac{(12)!}{(3)!(4)!(5)!}=27720
$

Hence, the answer is 27720

Example 4: The number of ways in which 6 distinct objects can be kept in two identical boxes so that no box remains empty is

Solution: Initially consider the boxes $B_1$ and $B_2$ to be distinct.
Object ' a ' can be kept in either of the boxes in 2 ways, similarly for all other things $\Rightarrow$ Total ways $=2^6$
But this includes when all the things are in $B_1$ or $B_2 \quad \Rightarrow$ Number of ways $=2^6-2$
Since the boxes are identical, therefore,

$
\frac{2^6-2}{2}=2^5-1=31
$

Hence, the answer is 31

Example 5: There are 9 chairs placed along a row. If there are 5 girls and 4 boys, then in how many ways can they sit such that all girls sit together and all boys sit together?

Solution: Put all the girls in one box and the boys in another box 5G 4B These two boxes can be arranged in 2 ! ways. In the first box, the 5 girls will arrange themselves in 5 ! ways, and in the other box, the boys will rearrange themselves in 4! ways.
$\therefore$ Total number of ways $=2!4!5$ !
Hence, the answer is $2!4!5$ !


Frequently Asked Questions (FAQs)

1. How does the concept of "formation of groups" differ from simple permutations?
While simple permutations deal with arranging all elements in a set, formation of groups focuses on dividing elements into subsets. This process often involves selecting some elements to form groups while leaving others out, or distributing all elements across multiple groups.
2. What is meant by the "formation of groups" in permutations and combinations?
Formation of groups refers to the process of arranging or selecting objects or people into distinct sets or clusters. In permutations and combinations, it involves organizing elements into smaller subsets based on specific criteria or rules.
3. What is the significance of "distinguishable" and "indistinguishable" objects in group formation?
Distinguishable objects are unique and can be told apart, while indistinguishable objects are identical. This distinction affects how we count arrangements and form groups, as permutations of indistinguishable objects don't create new arrangements.
4. How does the principle of addition apply to the formation of groups?
The principle of addition is used when we need to count the total number of ways to form groups in mutually exclusive cases. We add the number of ways for each case to get the total count.
5. When do we use the principle of multiplication in group formation problems?
We use the principle of multiplication when forming groups involves a series of independent choices or steps. The total number of ways to form groups is the product of the number of ways for each step.
6. What is the difference between "partition" and "distribution" in group formation?
Partition involves dividing a set of elements into distinct, non-overlapping groups. Distribution, on the other hand, involves placing elements into groups where the same element can appear in multiple groups or some groups may be empty.
7. How does the concept of "order matters" or "order doesn't matter" affect group formation?
When order matters (as in permutations), different arrangements of the same elements in a group are counted separately. When order doesn't matter (as in combinations), we only consider the selection of elements, not their arrangement within the group.
8. What is the role of "empty groups" in formation problems?
Some problems allow for empty groups, while others require all groups to contain at least one element. Allowing empty groups often increases the number of possible arrangements and can significantly change the problem-solving approach.
9. How do we handle problems involving the distribution of objects into distinct groups?
For distinct groups, we consider each group as a unique destination. We then use permutation or combination techniques to distribute objects, keeping in mind whether the objects are distinguishable and whether empty groups are allowed.
10. What is the difference between "distribution with repetition" and "distribution without repetition"?
Distribution with repetition allows an object to be placed in multiple groups or used multiple times. Distribution without repetition restricts each object to be used only once, placing it in a single group.
11. How does the concept of "derangement" relate to group formation?
A derangement is a permutation where no element appears in its original position. In group formation, this concept is useful when we need to distribute objects into groups ensuring that certain elements are not in specific positions or groups.
12. What is the "inclusion-exclusion principle" and how is it applied in group formation problems?
The inclusion-exclusion principle is used to count elements in the union of multiple sets without double-counting. In group formation, it's often applied when we need to count arrangements that satisfy multiple conditions or restrictions.
13. What is the role of "symmetry" in group formation problems?
Symmetry can simplify group formation problems by reducing the number of distinct arrangements. For example, in circular permutations or when dealing with identical objects, symmetry allows us to consider certain arrangements as equivalent.
14. What is the significance of "stirling numbers" in group formation problems?
Stirling numbers of the second kind count the number of ways to partition a set of n objects into k non-empty subsets. They are particularly useful in problems involving the distribution of distinguishable objects into indistinguishable groups.
15. What is the "pigeonhole principle" and how does it apply to group formation?
The pigeonhole principle states that if n items are placed into m containers, with n > m, then at least one container must contain more than one item. In group formation, this principle helps in proving the existence of certain arrangements or distributions.
16. What is the role of "recurrence relations" in solving group formation problems?
Recurrence relations are often used to solve complex group formation problems by breaking them down into smaller subproblems. They allow us to express the solution for n elements in terms of solutions for smaller numbers of elements.
17. How does the concept of "permutations with repetition" apply to group formation?
Permutations with repetition allow elements to be used multiple times in an arrangement. In group formation, this concept is useful when we can place the same element in multiple groups or use elements more than once within a group.
18. What is the significance of "Burnside's lemma" in group formation problems?
Burnside's lemma is used to count the number of orbits under a group action. In group formation, it's particularly useful for counting distinct arrangements when there are symmetries or rotations that make some arrangements equivalent.
19. What is the role of "Catalan numbers" in certain group formation problems?
Catalan numbers appear in various counting problems, including certain types of group formations. They're particularly relevant in problems involving balanced parentheses, which can be seen as a type of group formation with specific rules.
20. What is the "principle of inclusion-exclusion" and how is it applied in complex group formation problems?
The principle of inclusion-exclusion is a counting technique used to find the number of elements in the union of multiple sets. In complex group formation problems, it's often used to count arrangements that satisfy multiple conditions or avoid multiple restrictions.
21. How do we approach problems involving the formation of groups with elements that have weights or values?
For problems where elements have weights or values, we often use dynamic programming or generating functions. We consider both the number of elements and their total weight or value when forming groups.
22. What is the significance of "Stirling numbers of the first kind" in group formation?
Stirling numbers of the first kind count the number of permutations of n elements with k disjoint cycles. In group formation, they can be useful in problems involving circular arrangements or certain types of cyclic distributions.
23. What is the role of "generating functions" in solving complex group formation problems?
Generating functions are formal power series that encode sequences of numbers. In complex group formation problems, they're often used to derive recurrence relations, solve counting problems with multiple constraints, or handle problems involving distributions with specific patterns.
24. How do we approach problems involving the formation of groups with elements that have multiple, interdependent attributes?
For elements with interdependent attributes, we often need to use more advanced combinatorial techniques. This might involve creating a matrix or graph to represent the relationships between attributes, then using methods like the matrix-tree theorem or more sophisticated counting arguments.
25. What is the role of "graph theory" in solving certain group formation problems?
Graph theory can be useful in group formation problems, especially when there are complex relationships or constraints between elements. Concepts like bipartite matching, network flows, or coloring problems can often be applied to solve group formation problems with specific restrictions.
26. What is the significance of "Pólya enumeration theorem" in group formation problems involving symmetry?
The Pólya enumeration theorem is a powerful tool for counting orbits under a group action. In group formation problems involving symmetry, it can be used to count distinct arrangements when certain transformations (like rotations or reflections) produce equivalent configurations.
27. How does the concept of "circular permutation" relate to group formation?
Circular permutation is a special case of group formation where elements are arranged in a circle. In this case, rotations of the same arrangement are considered identical, reducing the total number of distinct arrangements.
28. What is the "stars and bars" technique in group formation?
The stars and bars technique is used to solve problems involving the distribution of indistinguishable objects into distinguishable groups. It represents objects as stars and group separations as bars, allowing for a visual approach to counting distributions.
29. How do we approach problems involving the formation of groups with restrictions?
When forming groups with restrictions (e.g., minimum or maximum group size, specific elements must be together or separate), we often use the complementary counting method or break the problem into cases that satisfy the given conditions.
30. What is the significance of the binomial coefficient in group formation?
The binomial coefficient (nCr) represents the number of ways to choose r objects from n objects without repetition and where order doesn't matter. It's crucial in many group formation problems, especially those involving combinations.
31. How do we handle problems involving the formation of groups with a fixed sum?
For problems where groups must have a fixed sum (e.g., distributing a certain number of objects into groups with specific totals), we often use generating functions or recursive methods to count the number of valid distributions.
32. How does the concept of "partitions" in number theory relate to group formation?
In number theory, partitions refer to ways of expressing a number as a sum of positive integers. This concept directly relates to group formation problems where we need to distribute objects into groups with specific sizes or totals.
33. What is the difference between "ordered partitions" and "unordered partitions" in group formation?
Ordered partitions consider the order of the groups, while unordered partitions do not. For example, splitting 5 objects into groups of (2,2,1) and (1,2,2) would be considered different in ordered partitions but the same in unordered partitions.
34. How do we approach problems involving the formation of groups with elements having multiple attributes?
When elements have multiple attributes (e.g., color and shape), we often use the multiplication principle to consider each attribute separately and then combine the results. Sometimes, we may need to use more advanced techniques like generating functions.
35. How do we handle problems involving the formation of groups with a fixed number of elements in each group?
For problems where each group must have a specific number of elements, we often use combinations to select elements for each group. If the groups are distinguishable, we may also need to consider the order of group formation.
36. How do we approach problems involving the formation of groups with elements that have preferences or restrictions?
When elements have preferences or restrictions (e.g., certain elements can't be in the same group), we often use graph theory concepts like bipartite matching or more advanced techniques like Hall's marriage theorem to determine valid group formations.
37. What is the significance of "Bell numbers" in group formation problems?
Bell numbers count the number of ways to partition a set into non-empty subsets. They are particularly useful in problems involving the distribution of distinguishable objects into any number of indistinguishable groups.
38. How do we handle problems involving the formation of groups with a mix of distinguishable and indistinguishable elements?
For problems with both distinguishable and indistinguishable elements, we often break the problem into cases based on how the distinguishable elements are distributed. Then, we use appropriate counting techniques for each case and sum the results.
39. What is the "exponential generating function" and how is it used in group formation problems?
An exponential generating function is a power series that encodes a sequence of numbers. In group formation, it's often used to solve complex counting problems, especially those involving distinguishable objects or where the order of selection matters.
40. How does the concept of "compositions" in number theory relate to group formation?
Compositions are ordered partitions of a number. In group formation, they relate to problems where we need to distribute objects into a fixed number of groups, considering the order of the groups and allowing empty groups.
41. What is the "multinomial coefficient" and how is it applied in group formation?
The multinomial coefficient generalizes the binomial coefficient to multiple categories. It's used in group formation problems where we need to distribute distinguishable objects into a fixed number of groups with specified sizes.
42. How do we approach problems involving the formation of groups with elements that must be kept together or separated?
For problems with elements that must be kept together or separated, we often treat the grouped elements as a single unit or use the complementary counting method to count arrangements that violate the conditions and subtract from the total.
43. How do we handle problems involving the formation of groups with a maximum capacity for each group?
For problems with maximum group capacities, we often use dynamic programming or generating functions. We consider the ways to fill each group without exceeding its capacity, then combine these results for all groups.
44. How does the concept of "derangement" extend to partial derangements in group formation?
Partial derangements allow some elements to remain in their original positions. In group formation, this concept is useful when we need to distribute objects into groups where some, but not all, objects must be moved from their initial positions.
45. How do we handle problems involving the formation of groups where the order of formation matters?
When the order of group formation matters, we often use the multiplication principle combined with permutations or combinations. We consider the ways to form each group in sequence, multiplying the number of choices for each step.
46. How does the concept of "partitions of a set" differ from "partitions of an integer" in group formation?
Partitions of a set involve dividing elements into non-empty subsets, while partitions of an integer involve expressing a number as a sum of positive integers. Both concepts are used in group formation, but set partitions deal with distinguishable elements, while integer partitions often relate to indistinguishable objects.
47. What is the "twelvefold way" and how does it relate to group formation problems?
The twelvefold way is a systematic classification of 12 related counting problems involving distributing balls into bins. It provides a framework for understanding various group formation problems based on whether the objects and groups are distinguishable or indistinguishable, and whether repetitions are allowed.
48. What is the significance of "Lah numbers" in certain group formation problems?
Lah numbers count the number of ways to partition a set into ordered subsets. They are particularly useful in problems involving the distribution of distinguishable objects into ordered, non-empty groups.
49. How do we handle problems involving the formation of groups with elements that can be split or combined?
For problems where elements can be split or combined, we often use dynamic programming or generating functions. We consider the ways elements can be divided or merged, and how these operations affect the overall group formation process.
50. How does the concept of "permutation groups" relate to more advanced group formation problems?
Permutation groups study the symmetries of a set of objects. In advanced group formation problems, understanding permutation groups can help identify equivalent arrangements, simplify counting processes, and solve problems involving symmetries or invariance under certain transformations.

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