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 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 isare TRUE-

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Multinomial Expansion

Let S1= i, j,...

Question

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

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?

n_{1} = 1000

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n_{2} = 44

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n_{3} = 220

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\frac{n_{4}}{12} = 420

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Solution

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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}},

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

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

A unit vector in the plane of the vectors 2i + j + k, i - j + k and orthogonal to 5i + 2j + 6k is

\vec{a} = 2 \hat{i} - 3 \hat{j} - \hat{k} and \vec{b} = \hat{i} + 4 \hat{j} - 2 \hat{k}, then \vec{a} \times \vec{b}

10 \hat{i} + 2 \hat{j} + 11 \hat{k}

10 \hat{i} + 3 \hat{j} + 11 \hat{k}

10 \hat{i} - 3 \hat{j} + 11 \hat{k}

10 \hat{i} - 2 \hat{j} - 10 \hat{k}

\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to j \times \to k))))))))

∣ ∣ ∣ ∣ \to v ∣ ∣ ∣ ∣

\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to j \times \to k))))))))

| | \to v | |

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