The meaning of combination is selection. Suppose we want to select two objects from four distinct objects a, b, c, and d. This can be stated as a number of combinations of four different objects taken two at a time. Here we have six different combinations ab, ac, ad, bc, bd, cd. In other words, we can say that there are six ways in which we can select two objects from four distinct objects. In real life, we use combinations for making lottery numbers and selecting nominees for student council.
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In this article, we will cover the Introduction to Combinations. 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 Combination?
The meaning of combination is selection.
The notation of selecting r objects from n given object is . Let’s derive the value of , and its relation with permutation notation.
We can generalize this concept for r object to be selected from given n objects as
Where 0 ≤ r ≤ n, and r is a whole number.
Now we have the value of
Applications of Combinations
Let us take an example of selecting things from two or more different groups:
Out of 5 men and 6 women in how many ways can a committee of 5 members be selected such that at least 2 members are women?
Solution:
The following cases are possible for at least 2 women,
2 women + 3 men =
3 women + 2 men =
4 women + 1 men =
5 women =
So, the total number of ways
The number of selection of r objects from n different objects:
This can be comprehended as taking out those k things that have to be included which can be done in 1 way and then finding the ways in which r-k objects can be selected from remaining n - k things, and putting those k things (which are already taken out) in r-k selected objects.
This can be comprehended as taking out k things that are not to be selected which can be done in 1 way and then finding the ways of selecting r things from n-k things.
This can be comprehended as taking out the q objects that should not be selected and putting it out and then taking out p objects that have to be selected and then finding ways of selecting r-p objects out of n- p-q objects and putting back p objects in r-p selected objects.
Example: In how many ways a cricket team can be selected out of 16 players such that 5 certain players must be included in the team?
Solution: Since 5 certain players have to be included so we need to select 11-5 = 6 players from 16 - 5 = 11 players.
So we can select the team in
If there are n points in the plane and out of which no three are collinear then,
Total No. of lines that can be formed using these n points = nC2
Total No. of triangles that can be formed using these n points = nC3
Total no. of Diagonals that can be formed in n-sided polygon = nC2 - n
If there are n points in the plane and out of which m points are collinear, then,
The total No. of different lines that can be formed by joining these n points is
The total No. of different triangles that can be formed by joining these n points is
The total No. of different quadrilaterals formed by joining these n points is
If m parallel lines in a plane are intersected by the family of other n parallel lines, then the total number of parallelograms formed is
The number of rectangles of any size in a square of size n x n is and number of squares of any size is .
In a rectangle of size n x p (n < p) number of rectangles of any size is .
To determine the number of ways to reach in the shortest way from point A to B.
When considering the possible paths or shortest path one can observe that the total number of steps in the forward direction is 6-R(Right) and in the upward direction is 4-U(Upward)
Now, If we arrange these 6 Rs and 4 Us in any way, it comes out to be the shortest path.
Or one can say that first find all the possible steps and arrange them to get the total number of possible ways.
Using "u" and "r" we can write out a path:
r r r r r r u u u u
r r r u u u u r r r
and others......
Hence, the total number of ways is or,
Solution: For the bijective function
First, select 17 images for these 17 inputs in ways and divide them in 1 way.
Now, select images for the rest of the 33 inputs in 33! ways.
Hence, the answer is
Example 2: Five numbers are randomly selected from the numbers and are arranged in increasing order . The probability that is? [JEE MAINS 2022]
Solution:
,
Hence, the answer is 1/ 68
Example 3: The sum of all the elements of the set is: [JEE MAINS 2022]
Solution
Hence, the answer is 1633.
Example 4: There are ten boys and five girls in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both together should not be the members of a group, is_____________. [JEE MAINS 2022]
Solution:
The number of ways of forming a group of 3 girls and 3 boys
The number of ways where two particular boys be members of a group together
Number of ways boys are not in the same group together
Required no.of selections =
Hence, the answer is .
Example 5: Let be 15 points on a circle. The number of distinct triangles formed by points , is : [JEE MAINS 2021a]
Solution: Total no. of Triangles
.....
These are 12 cases
So Desired no. of triangles
Hence, the answer is 443
Understanding the principles of combinations equips individuals with powerful tools to solve complex problems involving selection, grouping, and probability calculations efficiently. These applications underscore the fundamental role of combinations in both theoretical and practical domains. Knowledge of combinations is required for data analysis, algorithm design, or decision-making processes that involve combinatorial reasoning.
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