10 questions
PARAGRAPH – 1
Let \( S = \{1, 2, 3, 4, 5, 6\} \) and \( X \) be the set of all relations \( R \) from \( S \) to \( S \) that satisfy both the following properties:
i. \( R \) has exactly \( 6 \) elements.
ii. For each \( (a, b) \in R \), we have \( |a-b| \ge 2 \).
Let \( Y = \{\, R \in X : \text{ The range of } R \text{ has exactly one element} \,\} \) and \( Z = \{\, R \in X : R \text{ is a function from } S \text{ to } S \,\} \).
Let \( n(A) \) denote the number of elements in a set \( A \).
If $n(X) = {}^m C_6$, then the value of $m$ is ______________.
Let \( S \) be the set of all seven-digit numbers that can be formed using the digits \( 0 \), \( 1 \) and \( 2 \). For example, \( 2210222 \) is in \( S \), but \( 0210222 \) is NOT in \( S \). Then the number of elements \( x \) in \( S \) such that at least one of the digits \( 0 \) and \( 1 \) appears exactly twice in \( x \), is equal to ___________.
Let
\( S_1 = \{(i,j,k) : i,j,k \in \{1,2,\ldots,10\}\} \),
\( S_2 = \{(i,j) : 1 \le i < \; j+2 \le 10,\; i,j \in \{1,2,\ldots,10\}\} \),
\( S_3 = \{(i,j,k,l) : 1 \le i < j < k < l,\; i,j,k,l \in \{1,2,\ldots,10\}\} \) and
\( S_4 = \{(i,j,k,l) : i,j,k, \text{and} \, l \text{ are distinct elements in } \{1,2,\ldots,10\}\} .\)
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?
PARAGRAPH – 1
Let \( S = \{1, 2, 3, 4, 5, 6\} \) and \( X \) be the set of all relations \( R \) from \( S \) to \( S \) that satisfy both the following properties:
i. \( R \) has exactly \( 6 \) elements.
ii. For each \( (a, b) \in R \), we have \( |a-b| \ge 2 \).
Let \( Y = \{\, R \in X : \text{ The range of } R \text{ has exactly one element} \,\} \) and \( Z = \{\, R \in X : R \text{ is a function from } S \text{ to } S \,\} \).
Let \( n(A) \) denote the number of elements in a set \( A \).
If the value of \( n(Y) + n(Z) \) is \( k^{2} \), then \( |k| \) is __________.
Let \( \mathbb{N} \) denote the set of all positive integers. Consider the sets \( A = \{ 1, 2, 3, 4, 5 \} \) and \( B = \{ 1, 2, 3, 4, 5, 6, 7 \} \).
Let \( S \) be the set of all functions \( f : A \to B \) such that \( f(2) \neq 2 \) and \( f(4) \neq 4 \). Consider the set \( T = \{ f \in S : \text{ there exists a function } g : B \to \mathbb{N} \text{ such that } g(f(x)) = 2^x \text{ for all } x \in A \} \).
Then the number of elements in the set \( T \) is ________.
The number of \( 4 \)-digit integers in the closed interval \( [2022,4482] \) formed by using the digits \( 0,2,3,4,6,7 \) is ____________.
Let \( S = \{ 1, 2, 3, \ldots, 10 \} \). Consider the set \( X = \{ R : R \) is an equivalence relation on the set \( S \) such that \( R \) has exactly \( 42 \) elements\( \} \). Then the number of elements in \( X \) is ____________.
The number of ways to distribute \( 10 \) identical red pens and \( 14 \) identical blue pens among four persons such that each person gets \( 6 \) pens, is __________.
Consider \( 4 \) boxes, where each box contains \( 3 \) red balls and \( 2 \) blue balls. Assume that all \( 20 \) balls are distinct. In how many different ways can \( 10 \) balls be chosen from these \( 4 \) boxes so that from each box at least one red ball and one blue ball are chosen?
A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X, Y$, and $Z$ of sizes 2, 3, and 4, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is ______________.
