Question:

(C) $P \cap Q = \phi$ is

(P) $\frac{\binom{2n}{n}}{4^n}$

(Q) $\frac{2^{2n} - \binom{2n}{n}}{2^{2n+1}}$

(R) $\frac{\binom{2n}{n+1}}{4^n}$

(S) $\left(\frac{3}{4}\right)^n$

(T) $\frac{\binom{2n}{n}}{4^{n-1}}$

Step-by-Step Solution

Key Concept: For disjoint sets, each element has exactly 3 valid states instead of 4.

For each element $a \in A$, it can be in one of four states: (1) in neither P nor Q, (2) in P only, (3) in Q only, (4) in both P and Q.For $P \cap Q = \phi$, no element can be in both. So each element has only 3 choices: not in P or Q, in P only, or in Q only.Number of favorable outcomes: $3^n$Probability: $\frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n$

Correct Answer: S

Question:

Step-by-Step Solution

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