Question

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?

• A. $n_{1}=1000$
• B. $n_{2}=44$
• C. $n_{3}=220$
• D. $\frac{ n _{4}}{12}=420$

Answer 1

Answer: (A), (B), (D)

$n_{1}=1000$

Answer 2

Number of elements in S1 = 10 × 10 × 10 = 1000

Number of elements in S2 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 44

Number of elements in S3 = 10C4 = 210

Number of elements in S4 = 10P4 = 210 × 4! = 5040