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Standard XII

Mathematics

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

192

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408

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312

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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

Q. Let

(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

such that

1 \leq j < k \leq 2011

and

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

Q. Let

n \geq 3

be an integer. For a permutation

σ = (a_{1}, a_{2}, . . . . . . a_{n})

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

we let of

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

. Let

S_{σ}

be the sum of the roots of

f_{σ} (x) = 0

and let S denote the sum over all permutation

σ

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

of the number

S_{σ}

. Then.

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})

(1, 2, \dots, n)

define

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

Prove that there exist permutations

a \neq b

{1, 2, \dots, n}

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

Q. Let

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

be in G.P. with

a_{1} > 0

for

i = 1, 2, . . .10

and

S

be the set of pairs

(r, k), r k \in N

(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

Then the number of elements in

S

, is :

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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Let A be the set of all permutations a1,a2,....,a6 of 1,2,...,6 such that a1,a2,....ak is not a permutation of 1,2,...,k for any k,1≤k≤5. Then the number of elements in A is

The correct option is D 528Since ak 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!− 6C1×25=528

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

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 $6! -^{6} C_{1} \times 2^{5} = 528$