Let $S_1 = \{(i,j,k) : i, j, k \in \{1,2,\ldots,10\}\}$ $S_2 = \{(i,j) : 1 \le i < j + 2 \le 10, i, j \in \{1,2,\ldots,10\}\}$, $S_3 = \{(i,j,k,l) : 1 \le i < j < k < l, i, j, k, l \in \{1,2,\ldots,10\}\}$, and $S_4 = \{(i,j,k,l) : i, j, k \text{ and } l \text{ are distinct elements in } \{1,2,\ldots,10\}\}$. 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?

Let S₁ = {(i,j,k) : i, j, k ∈ {1,2,…,10}} S₂ = {(i,j) : 1 ≤ i < j + 2 ≤ 10, i, j ∈ {1,2,…,10}}, S₃ = {(i,j,k,l) : 1 ≤ i < j < k < l, i, j, k, l ∈ {1,2,…,10}}, and S₄ = {(i,j,k,l) : i, j, k and l are distinct elements in {1,2,…,10}}. If the total number of elements in the set Sᵣ is nᵣ, r = 1, 2, 3, 4, then which of the following statements is (are) TRUE?

$S_1 = \{(i,j,k) : i, j, k \in \{1,2,\ldots,10\}\}$

$S_2 = \{(i,j) : 1 \le i < j + 2 \le 10,\ i, j \in \{1,2,\ldots,10\}\}$

$S_3 = \{(i,j,k,l) : 1 \le i < j < k < l,\ i, j, k, l \in \{1,2,\ldots,10\}\}$

$S_4 = \{(i,j,k,l) : i, j, k, l \text{ distinct elements of } \{1,2,\ldots,10\}\}$

$\text{Find } n_1, n_2, n_3, n_4 \text{ and verify the statements } (1)\text{--}(4).$

$n_1 = 10 \times 10 \times 10 = 10^3 = 1000$

$\Rightarrow \text{statement } (1) \text{ is TRUE}$

$S_2: \ 1 \le i < j + 2 \le 10$

$j + 2 \le 10 \ \Rightarrow \ j \le 8$

$i < j + 2 \ \Rightarrow \ i \le j + 1$

$n_2 = \sum_{j=1}^{8} (j+1) = 2 + 3 + \cdots + 9$

$= 44 \ \Rightarrow \text{statement } (2) \text{ is TRUE}$

$\text{All pairs with } j \le 8: \ 10 \times 8 = 80$

$\text{Bad } (i \ge j+2): \ \sum_{j=1}^{8}(9-j) = 36$

$n_2 = 80 - 36 = 44 \ \checkmark$

$n_3 = \binom{10}{4} = \frac{10\cdot 9\cdot 8\cdot 7}{24} = 210$

$210 \neq 220 \ \Rightarrow \text{statement } (3) \text{ is FALSE}$

$n_4 = {}^{10}P_4 = 10\cdot 9\cdot 8\cdot 7 = 5040$

$\frac{n_4}{12} = \frac{5040}{12} = 420 \ \Rightarrow (4) \text{ TRUE}$

$n_1=1000,\quad n_2=44,\quad n_3=210,\quad n_4=5040$

$\text{TRUE statements: } (1),\ (2),\ (4)$