Let $$ \begin{gathered} \mathrm{S}_{1}=\{(\mathrm{i}, \mathrm{j}, \mathrm{k}): \mathrm{i}, \mathrm{j}, \mathrm{k} \in\{1,2, \ldots, 10\}\} \\ \mathrm{S}_{2}=\{(\mathrm{i}, \mathrm{j}): 1 \leq \mathrm{i}<\mathrm{j}+2 \leq 10, \mathrm{i}, \mathrm{j} \in\{1,2, \ldots, 10\}\}, \\ \mathrm{S}_{3}=\{(\mathrm{i}, \mathrm{j}, \mathrm{k}, l): 1 \leq \mathrm{i}<\mathrm{j}<\mathrm{k}<l, \mathrm{i}, \mathrm{j}, \mathrm{k}, l \in\{1,2, \ldots, 10\}\}. \end{gathered} $$ and $$ \mathrm{S}_{4}=\{(\mathrm{i}, \mathrm{j}, \mathrm{k}, l): \mathrm{i}, \mathrm{j}, \mathrm{k} \text { and } l \text { are distinct elements in }\{1,2, \ldots, 10\}\}. $$ If the total number of elements in the set $\mathrm{S}_{\mathrm{r}}$ is $\mathrm{n}_{\mathrm{r}}, \mathrm{r}=1,2,3,4$, then which of the following statements is (are) TRUE?

(A) $\mathrm{n}_{1}=1000$; (B) $\mathrm{n}_{2}=44$; (D) $\frac{\mathrm{n}_{4}}{12}=420$

JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Permutations and Combinations

Let

S_{1} = {(i, j, k) : i, j, k \in {1, 2, \dots, 10}} S_{2} = {(i, j) : 1 \leq i < j + 2 \leq 10, i, j \in {1, 2, \dots, 10}}, S_{3} = {(i, j, k, l) : 1 \leq i < j < k < l, i, j, k, l \in {1, 2, \dots, 10}} .

and

S_{4} = {(i, j, k, l) : i, j, k and l are distinct elements in {1, 2, \dots, 10}} .

If the total number of elements in the set

S_{r}

n_{r}, r = 1, 2, 3, 4

, then which of the following statements is (are) TRUE?

(A) $n_{1} = 1000$ ✓ Correct
(B) $n_{2} = 44$ ✓ Correct
(C) $n_{3} = 220$
(D) $\frac{n _{4}}{12} = 420$ ✓ Correct

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

Solution

For

S_{1}

: Each of i, j, k can independently take 10 values, so

n_{1} = 10 \times 10 \times 10 = 1000

. (A) is correct.

For

S_{2}

: The condition

1 \leq i < j + 2 \leq 10

means

j \leq 8

and

i \geq 1

. For each value of i from 1 to 9, we count valid j values. For i=1,2: j can be 1,2,...,8 (8 values each, total 16). For i=3: j can be 2,...,8 (7 values). Continuing this pattern and summing:

8 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 44

. (B) is correct.

For

S_{3}

: We need to choose 4 distinct numbers from {1,2,...,10} in strictly increasing order. This is simply

(4 10) = 210

, not 220. (C) is incorrect.

For

S_{4}

: Choose 4 distinct elements and arrange them in any order:

(4 10) \times 4! = 210 \times 24 = 5040

. Thus

\frac{n _{4}}{12} = \frac{5040}{12} = 420

. (D) is correct.

Correct answers: (A), (B), (D).

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JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Permutations and Combinations

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