JEE Advance - Mathematics (2021 - Paper 2 Online - No. 1)
Let
$${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$$,
$${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}$$,
$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \right\}$$ and
$${S_4} = \{ (i,j,k,l):i,j,k$$ 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?
$${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$$,
$${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}$$,
$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \right\}$$ and
$${S_4} = \{ (i,j,k,l):i,j,k$$ 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?
n1 = 1000
n2 = 44
n3 = 220
$${{{n_4}} \over {12}} = 420$$
Explanation
n1 = number of elements in S1 = 10 $$\times$$ 10 $$\times$$ 10 = 1000
n2 = number of elements in S2 = 8 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
= 8 + $${{8 \times 9} \over 2} = 44$$ ($$\because$$ 1 $$\le$$ i < j + 2 $$\le$$ 10 $$\Rightarrow$$ j $$\le$$ 8 and i $$\ge$$ 1
$$\therefore$$ (i = 1, j = 1, 2, 3, .... 8), (i = 2, j = 1, 2, 3, ..... 8), (i = 3, j = 2, 3, 4, .... 8) and so on)
n3 = number of elements in S3 = $${}^{10}{C_4}$$
(selecting 4 numbers and arranging in increasing order)
$$ = {{10 \times 9 \times 8 \times 7} \over {24}} = 210$$
n4 = number of elements in S4 = $${}^{10}{C_4}$$ = 10 $$\times$$ 9 $$\times$$ 8 $$\times$$ 7
$$ \Rightarrow {{{n_4}} \over {12}} = 420$$
n2 = number of elements in S2 = 8 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
= 8 + $${{8 \times 9} \over 2} = 44$$ ($$\because$$ 1 $$\le$$ i < j + 2 $$\le$$ 10 $$\Rightarrow$$ j $$\le$$ 8 and i $$\ge$$ 1
$$\therefore$$ (i = 1, j = 1, 2, 3, .... 8), (i = 2, j = 1, 2, 3, ..... 8), (i = 3, j = 2, 3, 4, .... 8) and so on)
n3 = number of elements in S3 = $${}^{10}{C_4}$$
(selecting 4 numbers and arranging in increasing order)
$$ = {{10 \times 9 \times 8 \times 7} \over {24}} = 210$$
n4 = number of elements in S4 = $${}^{10}{C_4}$$ = 10 $$\times$$ 9 $$\times$$ 8 $$\times$$ 7
$$ \Rightarrow {{{n_4}} \over {12}} = 420$$
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