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VOOZH | about |
Combinatorics is a branch of mathematics that deals with counting, arranging, and combining objects. Combinations are one of the fundamental topics in combinatorics, where we are required to select items from a set and the order of selection does not matter. But there may be additional restrictions or conditions on combinations which makes it difficult to find the solution.
These types of combination problems are generally known as restricted combinations. In this article, we will discuss what restricted combinations are, what types of restrictions can be there in combinations and also how to efficiently solve these kinds of problems.
Table of Content
A restricted combination is a type of combination where certain rules or limitations are applied to how items are selected from a group.
In a regular combination, we choose items without any specific conditions, but in a restricted combination, there are constraints, such as excluding certain items, requiring the inclusion of specific items, or limiting how many times an item can be chosen.
For example, if you're choosing 3 students from a group of 6, a restriction might be that one specific student must be included, or two specific students cannot be chosen together. These added conditions make the selection process more complex compared to an unrestricted combination.
Some examples of restricted combination are:
There are many different types of restrictions you can put on a restricted combination problem:
Restricted combinations can be tackled using several mathematical techniques,
Instead of directly counting the ways to satisfy the restriction, count the total number of combinations without restriction and subtract the invalid combinations.
This method is used when there are overlapping restrictions. You count the number of cases that violate one or more restrictions and adjust the total by adding or subtracting based on how many restrictions are violated simultaneously.
Break the problem into different cases based on the restriction, solve each case, and then sum the results.
Apply the restriction directly in the selection process by conditioning on a certain event or outcome. For instance, if one element must be included, select that element first, then count the remaining combinations.
Here is a comparison table highlighting the key differences between restricted and unrestricted combinations:
| Aspect | Restricted Combinations | Unrestricted Combinations |
|---|---|---|
| Definition | Combinations with certain limitations or conditions applied to how items are selected. | Combinations where no specific conditions or restrictions are applied to how items are selected. |
| Order of Selection | Order doesn't matter (as with all combinations), but there are restrictions such as the number of times an item can appear, or certain items must or must not be included. | Order doesn't matter, and items can be selected freely without any restrictions on their inclusion or repetition. |
| Repetition Allowed? | Sometimes. Repetition can be restricted (e.g., no repetition of elements). | Typically allows repetition of elements, depending on the context. For example, when selecting items with replacement. |
| Example of Usage | Selecting 3 cards from a deck where each suit is restricted to at most one card. | Selecting 3 cards from a deck with no restrictions on the number of suits. |
| Formula | Varies based on the restriction (e.g., may need specific counting methods like factorial adjustments). | Standard combination formula: C(n, r) = n!/[r!(n−r)!], where no restrictions apply. For combinations with repetition: C(n+r−1, r). |
| Complexity | Higher complexity due to imposed conditions or constraints. | Simpler, as no additional constraints need to be considered. |
| Real-World Example | Choosing a team of players with a specific number of forwards and defenders. | Choosing any 3 toppings for a pizza without any limit on which toppings can be chosen. |
| Mathematical Representation | Often requires custom formulas or constraints to be handled, depending on the restriction. | Standard combination formulas (with or without repetition). |
The key difference is that restricted combinations are subject to specific conditions or limitations, while unrestricted combinations are free from such constraints.
Let’s walk through some examples to solidify the concept:
Example 1: Choose 3 of the following fruits {apple, banana, orange, mango, grape} but you must not choose an orange.
Solution:
Firstly, determine the total number of what can be chosen from:
C(5,3) = 5!/3!(5-3)! = 10
After that, eliminate the invalid combinations in which an orange has been chosen. Since orange is excluded, we are left with {apple, banana, mango, grape}:
C(4,3) = 4!/3!(4-3)! = 4
So, literally it could be concluded that the number of the possibility of valid combination is four in total.
Example 2: Choose 3 fruit out of {apple, banana, orange, mango, grape} but you have to choose a banana.
Solution:
When a banana has to be included in the combination, choose it first. Now we need to select 2 more fruits from the remaining 4 fruits:
C(4,2) = 4!/2!(4-2)! = 6
Therefore it can be deduced that the total number of different and permissible combination that can be produced is six.
Example 3: Select 2 fruits out of Apple, Banana, Mango, Grape where, Mango and Apple cannot be selected together.
Solution:
Calculate total unrestricted combinations:
C(4,2) = 4!/2!(4-2)! = 6
Identify and subtract the invalid combinations where both Mango and Apple are chosen:
- Only one such combination: It contain {Mango, Apple}
Therefore, the total number of valid combinations: 6 - 1 = 5
Example 4: Select 3 fruits from a set that consist of 2 Apples, 3 Bananas and 1 Grape but it is allowed to take a maximum of 1 Apple.
Solution:
Case 1:Leave the number of apples required to buy zero.
- Choose 3 from {Banana, Banana, Banana, Grape}:
C(4,3) = 4!/3!(4-3)! = 4
Case 2: Choose One Apple.
- Choose 1 Apple and 2 from {Banana, Banana, Banana, Grape}:
C(2,1) × C(4,2) = 2 × 6 = 12
Total valid combinations:4 + 12 = 16
Example 5: We are given a group of 5 students {John, Alice, Bob, Carol, Dave} and we want to select 3 students such that at least John should be included In the selected team, if Alice will be included then Bob cannot be selected.
Solution:
Total unrestricted combinations:
C(5,3) = 10
Combinations without John:
- Choose 3 from {Alice, Bob, Carol, Dave}:
C(4,3) = 4
Valid combinations with at least one John: 10 − 4 = 6
Now account for the conditional restriction (if Alice is chosen, Bob cannot be chosen):Now account for the conditional restriction (if Alice is chosen, Bob cannot be chosen):
- Count combinations where both Alice and Bob are chosen along with John:
- Choose {John, Alice, Bob}: 1 blend, 1 mix.
- Subtract this invalid combination from the valid combinations: Subtract this invalid combination from the valid combinations: 6 − 1 = 5
Therefore the total number of valid combination must be equal five.
Problem 1: From a group of 6 students, select 3, but at least one student must be John.
Problem 2: Select 4 books from a collection of 8, but you cannot select any mystery books.
Problem 3: Form a team of 5 people from a group of 10, but if Alice is chosen, Bob cannot be selected.
Problem 4: Out of 52 cards in a deck, five cards are to be selected in way that is prohibited for any two among them belong to the same suit.
Problem 5: Taking it out for dinner, you are presented with a 10-course meal, out of which you must select four but two of those have to be vegetarian.
Problem 6: From 7 books, one wants to take 3 on his vacation, but he can’t choose the first and the second book at the same time.
Problem 7: Randomly pick five students from a class of twelve students with consideration being given to the seniority of the students.
Problem 8: Out of 8 colors, so paint a fence three of them could be selected seven of them cannot be red and blue simultaneously.
In conclusion, restricted combinations are a way to select items from a group while following specific rules or limitations. These restrictions can make the selection process more complex, but with the right methods, like complementary counting or case-by-case analysis, we can solve these problems effectively.
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