To determine which of the given statements are true, we need to calculate the number of elements, $n_r$, in each set $S_r$ for $r = 1, 2, 3, 4$.

Step 1: Calculate $n_1$ for set $S_1$

The set $S_1$ is defined as $S_1 = \{ (i,j,k) : i,j,k \in \{1, 2, ..., 10\} \}$. This is the set of all possible ordered triplets $(i,j,k)$ where each element can be any integer from 1 to 10.

• The number of choices for $i$ is 10.
• The number of choices for $j$ is 10.
• The number of choices for $k$ is 10. Since the choices are independent, the total number of elements in $S_1$ is the product of the number of choices for each component. $n_1 = 10 \times 10 \times 10 = 10^3 = 1000$

Checking Option A: The option states $n_1 = 1000$. This matches our calculation. Therefore, statement A is TRUE.

Step 2: Calculate $n_2$ for set $S_2$

The set $S_2$ is defined as $S_2 = \{ (i,j) : 1 \le i < j + 2 \le 10, i,j \in \{1, 2, ..., 10\} \}$. Let's analyze the conditions on $i$ and $j$.

1. $j+2 \le 10 \implies j \le 8$. Since $j \ge 1$, the possible values for $j$ are $j \in \{1, 2, 3, 4, 5, 6, 7, 8\}$.
2. $1 \le i < j+2$. The condition $1 \le i$ is already given. So we need to count pairs $(i,j)$ satisfying $i < j+2$ for the possible values of $j$, with $i \in \{1, ..., 10\}$.

We can count the number of possible values for $i$ for each value of $j$:

• If $j=1$: $i < 1+2=3$. Possible $i$ are $1, 2$. (2 pairs)
• If $j=2$: $i < 2+2=4$. Possible $i$ are $1, 2, 3$. (3 pairs)
• If $j=3$: $i < 3+2=5$. Possible $i$ are $1, 2, 3, 4$. (4 pairs)
• If $j=4$: $i < 4+2=6$. Possible $i$ are $1, ..., 5$. (5 pairs)
• If $j=5$: $i < 5+2=7$. Possible $i$ are $1, ..., 6$. (6 pairs)
• If $j=6$: $i < 6+2=8$. Possible $i$ are $1, ..., 7$. (7 pairs)
• If $j=7$: $i < 7+2=9$. Possible $i$ are $1, ..., 8$. (8 pairs)
• If $j=8$: $i < 8+2=10$. Possible $i$ are $1, ..., 9$. (9 pairs)

The total number of elements in $S_2$ is the sum of these counts: $n_2 = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9$ This is an arithmetic series. We can write it as $(\sum_{k=1}^9 k) - 1 = \frac{9(10)}{2} - 1 = 45 - 1 = 44$. $n_2 = 44$

Checking Option B: The option states $n_2 = 44$. This matches our calculation. Therefore, statement B is TRUE.

Step 3: Calculate $n_3$ for set $S_3$

The set $S_3$ is defined as $S_3 = \{ (i,j,k,l) : 1 \le i < j < k < l \le 10, i,j,k,l \in \{1, 2, ..., 10\} \}$. This is the set of all ordered 4-tuples of strictly increasing elements from the set $\{1, 2, ..., 10\}$. The number of such tuples is equal to the number of ways to choose 4 distinct elements from the set of 10 elements, because once the 4 elements are chosen, there is only one way to arrange them in increasing order. Thus, $n_3$ is the number of combinations of choosing 4 elements from 10, which is given by the binomial coefficient $\binom{10}{4}$. $n_3 = \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 10 \times 3 \times 7 = 210$

Checking Option C: The option states $n_3 = 220$. Our calculated value is $n_3 = 210$. Therefore, statement C is FALSE.

Step 4: Calculate $n_4$ for set $S_4$

The set $S_4$ is defined as $S_4 = \{ (i,j,k,l) : i,j,k \text{ and } l \text{ are distinct elements in } \{1, 2, ..., 10\} \}$. This is the set of all ordered 4-tuples of distinct elements from $\{1, 2, ..., 10\}$. This is equivalent to finding the number of permutations of 4 elements chosen from a set of 10 elements, denoted by $^{10}P_4$.

• Number of ways to choose the first element $i$ is 10.
• Number of ways to choose the second element $j$ (distinct from $i$) is 9.
• Number of ways to choose the third element $k$ (distinct from $i, j$) is 8.
• Number of ways to choose the fourth element $l$ (distinct from $i, j, k$) is 7. $n_4 = {^{10}P_4} = 10 \times 9 \times 8 \times 7 = 5040$

Checking Option D: The option states $\frac{n_4}{12} = 420$. Let's evaluate this expression using our calculated value of $n_4$. $\frac{n_4}{12} = \frac{5040}{12}$ Dividing 5040 by 12: $5040 \div 12 = 420$. So, the statement is $\frac{5040}{12} = 420$, which is true. Therefore, statement D is TRUE.

Conclusion

Based on our calculations:

• Statement A is TRUE ($n_1 = 1000$).
• Statement B is TRUE ($n_2 = 44$).
• Statement C is FALSE ($n_3 = 210$, not 220).
• Statement D is TRUE ($\frac{n_4}{12} = 420$).

The correct statements are A, B, and D.