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,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) textbf TRUE ?

Mathematics · Permutations and Combinations

JEE Advanced 2021 — Paper 2 — Question 18

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,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) \textbf{TRUE}?

$\mathrm{n}_{1}=10 \times 10 \times 10=10^{3}$

$\mathrm{n}_{2}={ }^{8} \mathrm{C}_{2}+2{ }^{8} \mathrm{C}_{1}=44$

$\mathrm{n}_{3}={ }^{10} \mathrm{C}_{4}=210$

$\mathrm{n}_{4}={ }^{10} \mathrm{P}_{4}=5040$

$\frac{\mathrm{n}_{4}}{12}=420$

Answer key and solution verified before publishing.

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