Question:
The triplet $(x, y, z)$ is chosen from the set $\{1,2,3,... n\}$, such that $x \le y < z$. The number of such triplets is
The triplet $(x, y, z)$ is chosen from the set $\{1,2,3,... n\}$, such that $x \le y < z$. The number of such triplets is
Updated On: Jul 7, 2022
- $ n^{3}$
- $^{n}C_{3}$
- $^{n}C_{2}$
- $^{n}C_{2} + \,^{n}C_{3}$
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The Correct Option is D
Solution and Explanation
Number of selections when $x < y < z$ is $^{n}C_{3}$.
Number of selections when $x = y < z$ is $^{n}C_{2}$.
$\therefore$ Required number $ = \,^{n}C_{3} + \,^{n}C_{2}$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".