Permutations and Combinations Test 28

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Permutations and Combinations Test 28

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Question 1
1 / -0

A question paper on mathematics consists of twelve questions divided into three parts A, B and C, each containing four questions. In how many ways can an examinee answer five questions, selecting atleast one from each part.

A
$$624$$

B
$$208$$

C
$$2304$$

D
None

Solution

Total no. of ways
$$A$$ $$B$$ $$ C$$ $$A$$ $$B$$ $$ C$$
$$=(^4C_1\times \ ^4C_1 \times \ ^4C_3)\times 3+ (^4C_1\, ^4C_2\, ^4C_2)\times 3$$ ($$\therefore$$ multiple by 3 as for each A,B,C different combination)
$$=192+432$$
$$=624$$
Question 2
1 / -0

There are $$k$$ different books and $$l$$ copies of each in a college library. The number of ways in which a student can make a selection of one or more books is

A
$$(k+1)^{l}$$

B
$$(l+1)^{k}$$

C
$$(k+1)^{l}-1$$

D
$$(l+1)^{k}-1$$

Solution
Question 3
1 / -0

$$7$$ boys and $$8$$ girls have to sit in a row on $$15$$ chairs numbered from $$1$$ to $$15$$ then?

A
Number of ways boys and girls sit alternately is $$8!7!$$

B
Number of ways boys and girls sit alternately is $$2(8!7!)$$

C
The number of ways in which first and fifteenth chair are occupied by boys and between any two boys an even number of girls sit is $$^9C_4$$ $$8!7!$$

D
The number of ways in which first and last seat are occupied by boys and between any two boys an even number of girls sit is $$(2{^9C_4}8!7!)$$.

Solution

$$\begin{matrix} G-denotes\, \, Girls \\ X-denotes\, \, boys \\ \Rightarrow G\times G\times G\times G\times G\times G\times G\times G \\ \Rightarrow Number\, \, of\, \, ways\, \, of\, Girls\, \, is\, \, 8!\, \, alternate \\ \Rightarrow Number\, \, of\, \, ways\, \, of\, boys\, \, is\, \, 7! \\ \, \, \, \, \, \, \, after\, \, late \\ \Rightarrow Number\, \, of\, \, boys\, \, and\, \, girls\, \, sit\, \, alternate\, \, =8!7! \\ When\, \, first\, \, and\, \, fifteen\, \, chair\, \, are\, \, fixed\, \, and\, \\ \, first\, \, chair\, \, \times G\times G\times G\times G\times G\times G\times G\times fifteen\, \, chair\, \, between\, \, any\, \, two\, \, boys \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \underline { 9{ C_{ 4 } }8!7! } \, \, \, Ans. \\ \end{matrix}$$
Question 4
1 / -0

Number of different paths of shortest distance from A to B in the grid which do not pass through M.

A
$$462$$

B
$$262$$

C
$$442$$

D
$$^{11}C_5-[^6C_3\times^5C_2]$$

Solution

$$\displaystyle \dfrac { 6! }{ 3!3! } =\dfrac { 6.5.4 }{ 3.2.1 } =20$$

$$\displaystyle \dfrac {11!}{6! 5!} = \dfrac { 11.10.9.8.7. }{ 5.4.3.2.1 }$$

$$\displaystyle 11\times42 = 462$$
Question 5
1 / -0

A is a set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the element of P.A subset Q of A is again chosen. The number of way of choosing P and Q so that P Q =$$\phi $$ is :-

A
$$2^{2n}-^{2n}C_{n}$$

B
$$2^{n}$$

C
$$2^{n}-1$$

D
$$3^{n}$$

Solution

Let A = $${a_{1},a_{2},a_{3},....a_{n}}$$. For $$a_{1}$$ $$\epsilon $$ For $$a_{1} \epsilon $$A we have following choices:
(i) $$a_{1} \epsilon $$ P and $$a_{1} \epsilon $$ Q
(ii) $$a_{1} \epsilon $$P $$\notin $$
(iii) $$_{1}\notin P and a_{1}\epsilon Q$$
(Iv)$$_{1}\notin P and a_{1}\notin Q$$
Out of these only (Ii), and (iii) and (iv) imply $$a_{1}\notin P\cap Q $$ therefore, the number of ways in which none of $$a_{1},a_{2}.....a_{n}$$ belong $$P\cap Q$$ is $$3^{n}$$.
Question 6
1 / -0

A box contains 5 pairs of shoes. If 4 shoes are selected, then the number of ways in which exactly one pair of shoes obtained is :

A
120

B
140

C
160

D
180

Solution

We have,

A box contains pair of shoes $$=5$$

Selected pair of shoes $$=4$$

Then,

Exactly one pair of shoes obtained is

$$ {{=}^{5}}{{P}_{4}} $$

$$ =\dfrac{5!}{\left( 5-4 \right)!} $$

$$ =\dfrac{5!}{1!}=5! $$

$$ =5\times 4\times 3\times 2\times 1 $$

$$ =120 $$
Hence, this is the answer.
Question 7
1 / -0

There are n locks and n matching keys. If all the locks and keys are to be perfectly matched, find the maximum number of trails required to open a lock.

A
$$^{n}C_{2}$$

B
$$\sum _{ k=2 }^{ n }{ \left( k+2 \right) } $$

C
$$\dfrac{n(n+1)}{2}$$

D
$$^{n+1}C_{2}$$

Solution
Question 8
1 / -0

Given $$4$$ flags of different colours, how many different signals can be generated. If a signal requires the use of $$2$$ flags one below the other?

A
$$4$$

B
$$3$$

C
$$12$$

D
$$1$$

Solution

No. of ways of selecting two flags out of four = $$ ^4C_2$$

So, total possible different signals generated = $$^4C_2\times2!$$

$$\implies 6\times2=12$$
Question 9
1 / -0

There are $$4$$ letter boxes in a post office. In how many ways can a man post $$8$$ distinct letters?

A
$$4\times 8$$

B
$${8}^{4}$$

C
$${4}^{8}$$

D
$$P\left (8,4\right)$$

Solution

$$ \rightarrow $$ Each letter can go to 4 boxes
$$ \therefore $$ Total number of ways $$ = 4\times 8 = 32 $$
Question 10
1 / -0

$$4$$ buses runs between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, then the total possible ways are

A
$$12$$

B
$$16$$

C
$$4$$

D
$$8$$

Solution

$$ \rightarrow $$ On going Bhopal to Gwalior
he has 4 options

On returning he has

(4-1) options = 3

$$ \therefore $$ Total ways $$ = 4 \times3 $$ = 12 ways