Permutations and Combinations Test 18

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Permutations and Combinations Test 18

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Question 1
1 / -0

The total number of five digit numbers the sum of whose digits is even is

A
$$4.5\times {10}^4$$

B
$$9\times{10}^4$$

C
$${10}^5$$

D
none of these

Solution

$$0,1,2,3,4,5,6,7,8,9$$
Five digit number
Last digit can be filled in $$5$$ ways
That is if the first four digits add up to odd sum then last digit will be odd
Else it would be even
Total number of such digits
$$=9\times 10\times 10\times 10\times 5\\ =45\times 10^{ 3 }$$
$$=4.5\times 10^{ 4 }$$
Hence, the correct answer is $$4.5\times 10^{ 4 }$$.
Question 2
1 / -0

If $$^{40}C_{n+7} = ^{40}C_{4n-2},$$ then all the values of n are given by

A
$$28$$

B
$$3,6$$

C
$$3,7$$

D
$$6$$

Solution

Here there are two conditions
$$n+7=4n-2$$
$$\Longrightarrow 3n=9$$
$$\Longrightarrow n=3$$
and,
$$n+7+4n-2=40$$
$$\Longrightarrow 5n+5=40$$
$$5n=35$$
$$n=7$$
There are two possible value of $$n$$ . They are $$3$$ and $$7$$
Question 3
1 / -0

If, $$\dfrac{1}{^5C_r}+\dfrac{1}{^6C_r} = \dfrac{1}{^4C_r}$$, then the value of $$r$$ equals to

A
$$4$$

B
$$2$$

C
$$5$$

D
$$3$$

Solution

Since, there is a $$^4 C_r$$ in the expression, then $$r \le 4$$
$$\dfrac{1}{\:^{5}C_{r}}+\dfrac{1}{\:^{6}C_{r}}=\dfrac{1}{\:^{4}C_{r}}$$

$$\dfrac{\:^{6}C_{r}+\:^{5}C_{r}}{\:^{5}C_{r}.\:^{6}C_{r}}=\dfrac{1}{\:^{4}C_{r}}$$

$$\dfrac{\:^{5}C_{r}(\dfrac{2(6)-r}{6-r})}{\:^{5}C_{r}.\:^{6}C_{r}}=\dfrac{1}{\:^{4}C_{r}}$$

$$\dfrac{12-r}{6-r}=\dfrac{\:^{6}C_{r}}{\:^{4}C_{r}}$$

$$\dfrac{12-r}{6-r}=\dfrac{\dfrac{6!}{(6-r)!}}{\dfrac{4!}{(4-r)!}}$$

$$\dfrac{12-r}{6-r}=\dfrac{5\times 6}{(6-r)(5-r)}$$
$$(12-r)(5-r)=30$$
$$(12-r)(5-r)=10\times 3$$
$$12-r=10$$ or $$r=2$$.. similarly $$5-r=3$$ or $$r=2$$.
Hence $$r=2$$ is a solution.
Thus the answer is $$r=2$$.
Question 4
1 / -0

Ten different letters of alphabet are given, words with five letters are formed with these given letters. Then the number of words which have at least one letter repeated

A
$$69760$$

B
$$30240$$

C
$$99748$$

D
$$42386$$

Solution

$$10$$ letters $$5$$ to be selected
Total number of $$5$$ letter words $$={ 10 }^{ 5 }$$
Number of words without repetition $$={}^{ 10 }{ P }_{ 5 }$$
Number of words without at least one letter repeated
$$=100000-(10\times 9\times 8\times 7\times 6)\\ =69760 $$
Hence the correct answer is $$69760$$
Question 5
1 / -0

When listing the integers from $$1$$ to $$1000$$, how many times the digit $$5$$ be written?

A
$$297$$

B
$$243$$

C
$$300$$

D
$$273$$

Solution

Given range of integer $$(1 $$to$$ 1000)$$
From $$1$$to $$100$$
The number of $$5$$ in unit place $$=10$$
From $$(1 to 1000)$$ there will be $$100$$ $$5's$$ in unit place.
Similarly in $$100$$ place there will be $$10 $$ $$'5's'$$
From $$1 $$to$$ 1000$$ There wold be $$100$$ $$'5's'$$ in tens place
Also in $$100's$$ place There would be $$100$$ $$'5's'$$
Total Number of $$5's $$
$$\Rightarrow 100+100+100\\ \Rightarrow 300$$
Hence the correct answer is $$300$$
Question 6
1 / -0

If $$^nC_8 = ^nC_{27}$$, then what is the value of $$n?$$

A
$$35$$

B
$$22$$

C
$$28$$

D
$$41$$

Solution

Given,
$$\Rightarrow ^{ n }{ C }_{ 8 }=^{ n }{ C }_{ 27 }$$
We know ,
$$\Rightarrow ^{ n }{ C }_{ r }=^{ n }{ C }_{ n-r }\\ \Rightarrow r+(n-r)=n\\ \Rightarrow 8+27=n\\ \Rightarrow n=35\\ \\ $$
Option $$:A$$
Question 7
1 / -0

The exponent of $$18$$ in $$200!$$, is

A
$$24$$

B
$$46$$

C
$$47$$

D
$$48$$

Solution

Exponent of $$18$$ in $$200!$$

$$\Rightarrow 18=(3)^{ 2 }\times 2$$

Exponent of $$P$$ in $$n!$$

$$=\left[ \dfrac { n }{ p } \right] +\left[ \dfrac { n }{ p^{ 2 } } \right] +.............\left[ \dfrac { n }{ p^{ K } } \right] \\ =P^{ K }\le n$$

Exponential of $$3$$ in $$n!$$

$$=\left[ \dfrac { 200 }{ 3 } \right] +\left[ \dfrac { 200 }{ 9 } \right] +\left[ \dfrac { 200 }{ 27 } \right] +\left[ \dfrac { 200 }{ 81 } \right] \\ =66+22+4+2\\ =94$$

Exponent of $$(3)^{ 2 }=48$$

Exponent of $$2$$ in $$n!$$

$$=\left[ \dfrac { 200 }{ 2 } \right] +\left[ \dfrac { 200 }{ 4 } \right] +\left[ \dfrac { 200 }{ 8 } \right] +\left[ \dfrac { 200 }{ 16 } \right] >>48$$

Least exponent $$=48$$

Exponential of $$18$$ is $$48$$

Hence the correct answer is $$48$$
Question 8
1 / -0

Find the number of subsets of the set $${1,2,3,4,5,6,7,8,9,10,11}$$ having $$4$$ elements

A
$$330$$

B
$$320$$

C
$$310$$

D
$$300$$

Solution

Number of digits $$=11$$

$$1,2,3,4,5,6,7,8,9,10,11$$

$$4$$ Elements are to be chosen

Number of ways $$^{ 11 }{ C }_{ 4 }$$

$$=\cfrac { 11! }{ 4!\times 7! } $$

$$ =\cfrac { 11\times 10\times 9\times 8 }{ 4\times 3\times 2 } $$

$$ =330$$

Number of subsets $$=330$$
Question 9
1 / -0

There are $$8$$ true/false questions in an examination.The number of ways in which this questions can be answered, is

A
$$256$$

B
$$1024$$

C
$$16$$

D
$$64$$

Solution

$$8$$ True or False
Each questions has two choices
Two questions has $$2\times 2$$ choices
Similarly $$8$$ questions have $${ (2) }^{ 8 }$$ choices
$$\Rightarrow 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\\ \Rightarrow 4\times \times 4\times 4\times 4\\ \Rightarrow 16\times 16$$
$$\Rightarrow 256$$ Choices
Hence the correct answer is $$256$$
Question 10
1 / -0

There are $$8$$ men and $$10$$ women and you need to form a committee of $$5$$ men and $$6$$ women. In how many ways can the committee be formed?

A
$$10420$$

B
$$11420$$

C
$$11760$$

D
None of these

Solution

Out of $$8 $$ men and $$10$$ women . The committee has to select $$5$$ men and $$6$$ women.

$$^8C_5 \times ^{10}C_6 =11760$$

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Re-Attempt Weekly Quiz Competition

Permutations and Combinations Test 18

Result

Permutations and Combinations Test 18

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SHARING IS CARING

Question 1

$$4.5\times {10}^4$$

$$9\times{10}^4$$

$${10}^5$$

none of these

Solution

Question 2

$$28$$

$$3,6$$

$$3,7$$

$$6$$

Solution

Question 3

$$4$$

$$2$$

$$5$$

$$3$$

Solution

Question 4

$$69760$$

$$30240$$

$$99748$$

$$42386$$

Solution

Question 5

$$297$$

$$243$$

$$300$$

$$273$$

Solution

Question 6

$$35$$

$$22$$

$$28$$

$$41$$

Solution

Question 7

$$24$$

$$46$$

$$47$$

$$48$$

Solution

Question 8

$$330$$

$$320$$

$$310$$

$$300$$

Solution

Question 9

$$256$$

$$1024$$

$$16$$

$$64$$

Solution

Question 10

$$10420$$

$$11420$$

$$11760$$

None of these

Solution

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