Let \( X=\{1,2,3, \ldots \ldots, 10\} . A, B, C \) are three sets such that \( A \subseteq X, B \subseteq X \) and \( C \subseteq X \). Column-1: Contains types of three subsets of \( X \).
Column-2: Contains number of ways of selecting three subsets of \( X \) according to column-1
Column-3: Contains conditional probabilities \( P\left(\frac{E}{E_{1}}\right) \) or \( P\left(\frac{E}{E_{2}}\right) \) where
\( E \) : Selecting three subsets of \( X \) according to column-1
\( E_{1} \) : Selecting three subsets of \( X \) such that \( n(A \cap B)=5 \)
\( E_{2}: \) Selecting three subsets of \( X \) such that \( n(A \cup B)=5 \).
Column-1
Column-2
Column-3
(I) \( A \cap B \cap C \supseteq\{2,3,4,5,6\} \) and \( A=B=C \)
(i) 32
(P) \( \quad P\left(\frac{E}{E_{1}}\right)=0 \)
(II) \( A \cup B \cup C=\{3,4,5\} \)
(ii) 242
(Q) \( P\left(\frac{E}{E_{1}}\right)=\frac{1}{{ }^{10} C_{5} \cdot 12^{5}} \)
(III) \( A \cap B \cap C=\{3,4,5,6,7\} \) and \( A=B \neq C \)
(iii) 243
(R) \( P\left(\frac{E}{E_{2}}\right)=\frac{31}{{ }^{10} C_{5} \cdot 12} \)
(IV) \( A \cup B \cup C=\{6,7,8,9,10\} \) and \( A=B \neq C \)
(iv) 343
(S) \( P\left(\frac{E}{E_{2}}\right)=0 \)
[Note: \( S \supseteq T \) denotes \( S \) is a superset of \( T \), means \( S \) contains atleast all elements of \( T \)
646. Which of the following options is the only correct combination?
(a) (I) (i) (P)
(b) (II) (ii) (S)
(c) (III) (ii) (R)
(d) (IV) (iv) (P)
