1. Tardigrade
  2. Question
  3. Mathematics
  4. Let S1= ( i , j , k ): i , j , k ∈ 1,2, ldots ., 10 S2= ( i , j ): 1 ≤ i < j +2 ≤ 10, i , j ∈ 1,2, ldots . .10 S3= ( i , j , k , l ): 1 ≤ i < j < k < l , i , j , k , l ∈ 1,2, ldots ., 10 and S4= ( i , j , k , l ): i , j , k and l 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?

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Q. 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\}\}$ and
$ S_{4}=\{( i , j , k , l ): i , j , k $ and $ l$ 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?

JEE AdvancedJEE Advanced 2021

Solution:

$n _{1}=10 \times 10 \times 10=10^{3}$
$n _{2}={ }^{8} C _{2}+2{ }^{8} C _{1}=44$
$n _{3}={ }^{10} C _{4}=210$
$n _{4}={ }^{10} P _{4}=5040$