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Number of Functions Possible from Set A to Set B

Let A be th...

Question

Let $A$ be the set of all permutations $a_{1}, a_{2}, . . . ., a_{6}$ of $1, 2, . . ., 6$ such that $a_{1}, a_{2}, . . . . a_{k}$ is not a permutation of $1, 2, . . ., k$ for any $k, 1 \leq k \leq 5$ . Then the number of elements in $A$ is

A

192

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B

408

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C

312

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D

528

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Solution

The correct option is D $528$
Since $a_{k}$ does not have $k$ element as a permutation, it has $5$ elements each with 2 possibilities (one of being in the permutation and the other of not being the permutation). Hence, total permutations possible are $6! -^{6} C_{1} \times 2^{5} = 528$

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Similar questions

(a_{1}, a_{2}, a_{3}, . . ., a_{2011})

be a permutation (that is a rearrangement) of the numbers 1, 2, 3, . . . , 2011. Show that there exist two numbers

j, k

1 \leq j < k \leq 2011

| a_{j} - j | = | a_{k} - k |

n \geq 3

be an integer. For a permutation

σ = (a_{1}, a_{2}, . . . . . . a_{n})

(1, 2, . . . ., n)

σ (x) = a_{n} x^{n - 1} + a^{n - 1} + . . . . + a_{2} x + a_{1}

S_{σ}

be the sum of the roots of

f_{σ} (x) = 0

and let S denote the sum over all permutation

σ

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S_{σ}

Q. Let n be an odd integer greater than 1 and let

c_{1}, c_{2}, \dots, c_{n}

be integers. For each permutation

a = (a_{1}, a_{2}, \dots, a_{n})

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S (a) = n \sum i = 1 c_{i} a_{i} .

Prove that there exist permutations

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such that n! is a divisor of S(a)-S(b).

a_{1}, a_{2}, a_{3}, . . . ., a_{10}

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S

be the set of pairs

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(the set of natural numbers) for which

∣ ∣ ∣ ∣ ∣ \begin{matrix} {log}_{e} a_{1}^{r} a_{2}^{k} & {log}_{e} a_{2}^{r} a_{3}^{k} & {log}_{e} a_{3}^{r} a_{4}^{k} {log}_{e} a_{4}^{r} a_{5}^{k} & {log}_{e} a_{5}^{r} a_{6}^{k} & {log}_{e} a_{6}^{r} a_{7}^{k} {log}_{e} a_{7}^{r} a_{8}^{k} & {log}_{e} a_{8}^{r} a_{9}^{k} & {log}_{e} a_{9}^{r} a_{10}^{k} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

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A sequence $a_{1}, a_{2}, a_{3}, . . . . . . .$ is defined by letting $a_{1} = 3$ and $a_{k} = 7$ $a_{k - 1}$ for all natural numbers $k \geq 2$ . Show that $a_{n} = {3.7}^{n - 1}$ for all $n ϵ N$ .

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