Let
$S_{1}=\{( i , j , k ): i , j , k \in\{1,2, \ldots , 10\}\},$
$ S_{2}=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots 10\}\},$
$ S_{3}=\{( i , j , k , l ): 1 \leq i < j < k < l , i , j , k , l \in\{1,2, \ldots , 10\}\}$ and
$ S_{4}=\{( i , j , k , l ): i , j , k $ and $ l$ are distinct elements in $\{1,2, \ldots , 10\}\}$.
If the total number of elements in the set $S_{ r }$ is $n_{ r }, r =1,2,3,4$, then which of the following statements is(are) TRUE?
Let
$S_{1}=\{( i , j , k ): i , j , k \in\{1,2, \ldots , 10\}\},$
$ S_{2}=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots 10\}\},$
$ S_{3}=\{( i , j , k , l ): 1 \leq i < j < k < l , i , j , k , l \in\{1,2, \ldots , 10\}\}$ and
$ S_{4}=\{( i , j , k , l ): i , j , k $ and $ l$ are distinct elements in $\{1,2, \ldots , 10\}\}$.
If the total number of elements in the set $S_{ r }$ is $n_{ r }, r =1,2,3,4$, then which of the following statements is(are) TRUE?
- $n_{1}=1000$
- $n_{2}=44$
- $n_{3}=220$
- $\frac{ n _{4}}{12}=420$
The Correct Option is A, B, D
Solution and Explanation
Step 1: Number of elements in S1:
The set \( S_1 \) contains triples of the form \( (i, j, k) \), where \( i, j, k \in \{1, 2, \dots, 10\} \).
The number of choices for \( i \), \( j \), and \( k \) are 10 each.
Thus, the total number of elements in \( S_1 \) is:
\( n_1 = 10 \times 10 \times 10 = 1000 \).
Step 2: Number of elements in S2:
The set \( S_2 \) contains pairs of the form \( (i, j) \) where \( 1 ≤ i < j + 2 ≤ 10 \) and both \( i \) and \( j \) are in \( \{1, 2, \dots, 10\} \).
To find the total number of valid pairs \( (i, j) \), observe the following:
For each value of \( i \), the value of \( j \) can range from \( i + 1 \) to \( 10 \), subject to the constraint \( i + j + 2 ≤ 10 \).
Thus, for \( i = 1 \), the possible values of \( j \) are 2 through 9 (8 choices),
for \( i = 2 \), the possible values of \( j \) are 3 through 8 (7 choices), and so on.
Summing the choices gives the total number of elements in \( S_2 \):
\( 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 44 \).
Step 3: Number of elements in S3:
The set \( S_3 \) contains quadruples of the form \( (i, j, k, l) \), where \( 1 ≤ i < j < k < l ≤ 10 \).
This is equivalent to choosing 4 distinct elements from a set of 10 elements, which can be done in \( \binom{10}{4} \) ways.
Using the combination formula:
\( \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \).
Step 4: Number of elements in S4:
The set \( S_4 \) contains quadruples of the form \( (i, j, k, l) \), where \( i, j, k, l \) are distinct elements in \( \{1, 2, \dots, 10\} \).
This is equivalent to choosing 4 distinct elements from a set of 10, and then arranging them in order (since the order matters).
The number of ways to do this is given by the permutation formula:
\( P(10, 4) = \frac{10!}{(10-4)!} = 10 \times 9 \times 8 \times 7 = 5040 \).
Step 5: Final Answer:
The correct statements are:
A) \( n_1 = 1000 \)
B) \( n_2 = 44 \)
D) \( \frac{n_4}{12} = 420 \), since \( \frac{5040}{12} = 420 \).
Top Questions on sets
- If \( R \) is the smallest equivalence relation on the set \( \{1, 2, 3, 4\} \) such that \( \{(1,2), (1,3)\} \subseteq R \), then the number of elements in \( R \) is ______.
- If \( a>0, b>0, c>0 \) and \( a, b, c \) are distinct, then \( (a + b)(b + c)(c + a) \) is greater than:
- The modulus of the complex number \( z \) such that \( |z + 3 - i| = 1 \) and \( \arg(z) = \pi \) is equal to:
- In a statistical investigation of 1003 families of Calcutta, it was found that 63 families have neither a radio nor a TV, 794 families have a radio, and 187 have a TV. The number of families having both a radio and a TV is:
- Number of subsets of set of letters of word 'MONOTONE' is:
Questions Asked in JEE Advanced exam
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
- JEE Advanced - 2025
- Coordinate Geometry
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
- JEE Advanced - 2025
- Waves and Oscillations
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____.
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80- JEE Advanced - 2025
- Organic Chemistry
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- binomial expansion formula
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions
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 "|".