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Let
$S _1=\{( i , j , k ): i , j , k \in\{1,2, \ldots, 10\}\}$
$S _2=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots, 10\}\},$
$S _3=\{( i , j , k , l): 1 \leq i < j < k < l, i , j , k , l \in\{1,2, \ldots ., 10\}\}$
$S _4=\{( i , j , k , l): i , j , k$ and $l$ are distinct elements in $\{1,2, \ldots, 10\}\}$
and If the total number of elements in the set $S _t$ is $n _z, r =1,2,3,4$, then which of the following statements is (are) TRUE?
$(A)$ $n _1=1000$ $(B)$ $n _2=44$ $(C)$ $n _3=220$ $(D)$ $\frac{ n _4}{12}=420$
$A,B,C$
$A,B$
$A,B,D$
$A,C$
Solution
$(A)$ $n _1=10 \times 10 \times 10=1000$
$(B)$ As per given condition $1 \leq i < j +2 \leq 10 \Rightarrow j \leq 8$ & $i \geq 1$ for $i =1,2, \quad j =1,2,3, \ldots, 8 \rightarrow(8+8)$ possibilities for $i =3, \quad j =2,3, \ldots, 8 \rightarrow 7$ possibilities $i =4, \quad j =3, \ldots, 8 \rightarrow 6$ possibilities $i =9, \quad j =1 \quad \rightarrow 1$ possibility
So $n _2=(1+2+3+\ldots . .+8)+8=44$
$(C)$ $n _3={ }^{10} C _4$ (Choose any four)
$=210$
$(D)$ $n _4={ }^{10} C _4 \cdot 4!=(210)$
$\Rightarrow \frac{ n _4}{12}=420$
So correct Ans.$(A), (B), (D)$
