MCQ
$\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}$ is equal to
- ✓$2^{2 n }-{ }^{2 n } C _{ n }$
- B$2^{2 n -1}-^{2 n -1} C _{ n -1}$
- C$2^{2 n }-\frac{1}{2}{ }^{2 n } C _{ n }$
- D$2^{ n -1}+{ }^{2 n -1} C _{ n }$
$=\sum_{ i =0}^{ n }{ }^{ n } C _{ i } \cdot \sum_{ j =0}^{ n }{ }^{ n } C _{ j }-\sum_{ i = j =0}^{ n }\left({ }^{ n } C _{ i }\right)^{2}$
$=\left(2^{ n }\right)\left(2^{ n }\right)-{ }^{2 n } C _{ n }$
$=2^{2 n }-{ }^{2 n } C _{ n }$
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$S=\left\{X \in R ^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and }$
$T=\left\{Y \in R ^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\}.$
Then which of the following statements is (are) $TRUE$?
$(A)$ There is a triangle whose area is $1$ and all of whose vertices are from $S$.
$(B)$ There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
$(C)$ There are infinitely many rectangles of perimeter $48$ , two of whose vertices are from $S$ and the other two vertices are from $I$.
$(D)$ There is a square of perimeter $48$ , two of whose vertices are from $S$ and the other two vertices are from $T$.