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 \( \frac{5040}{12} = 420 \).
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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 "|".