Permutations and Combinations Test 25

Result

Permutations and Combinations Test 25

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

If $$\dfrac { ^{ n }{ P }_{ r-1 } }{ a } =\dfrac { ^{ n }{ P }_{ r } }{ b } =\dfrac { ^{ n }{ P }_{ r+1 } }{ c } $$, then which of the following holds good:

A
$$c^{2}=a(b+c)$$

B
$$a^{2}=c(a+b)$$

C
$$b^{2}=c(a-b)$$

D
$$\dfrac {1}{a}+\dfrac {1}{b}+\dfrac {1}{c}=1$$

Solution

$$\dfrac{^{n} {P}_{r-1}}{a} = \dfrac{^{n}{P}_{r}}{b} = \dfrac{^{n}{P}_{r+1}}{c} $$

$$\implies \dfrac{n!}{(n - r+ 1)!a} = \dfrac{n!}{(n-r)!b} = \dfrac{n!}{(n-r-1)!c} $$

$$\implies \dfrac{n!}{(n - r+ 1)!a} = \dfrac{n!}{(n-r)(n-r+1)!b} = \dfrac{n!}{(n-r-1)(n-r)(n-r+1)!c} $$

$$\implies \dfrac{1}{a} = \dfrac{1}{(n-r)b} = \dfrac{1}{(n-r-1)(n-r)c} $$

$$\implies n-r = \dfrac{a}{b}$$ and $$n-r-1 = \dfrac{b}{c}$$

$$\implies n-r - 1 = \dfrac{a}{b} - 1 $$

$$\implies \dfrac{b}{c} = \dfrac{a}{b} - 1$$

$$\implies \dfrac{b}{c} - \dfrac{a}{b} = - 1$$

$$\implies b^2 - ac = - bc $$

$$\implies b^2 = ac - bc$$

$$\therefore b^2 = c(a - b)$$ (Ans)
Question 2
1 / -0

If $$^nC_r : ^nC_{r + 1} : ^nC_{r + 2} = 3 : 4: 5$$, then the value of $$2n + 3r$$ is

A
$$238$$

B
$$220$$

C
$$203$$

D
$$240$$

Solution

$$\dfrac { { { n }_{ C } }_{ r } }{ { { n }_{ C } }_{ r+1 } } =\dfrac { 3 }{ 4 } $$

$$ \dfrac { r+1 }{ n-r } =\dfrac { 3 }{ 4 } $$

$$ \dfrac { { { n }_{ C } }_{ r+1 } }{ { { n }_{ C } }_{ r+2 } } =\dfrac { 4 }{ 5 } $$

$$ \dfrac { r+2 }{ n-r-1 } =\dfrac { 4 }{ 5 } $$

$$ On\quad solving\quad above\quad we\quad get\quad r=26,n=62$$

$$ 2n+3r=238$$

$$\dfrac { { { n }_{ C } }_{ r } }{ { { n }_{ C } }_{ r+1 } } =\dfrac { 3 }{ 4 } $$

$$ \dfrac { r+1 }{ n-r } =\dfrac { 3 }{ 4 } $$

$$ \dfrac { { { n }_{ C } }_{ r+1 } }{ { { n }_{ C } }_{ r+2 } } =\dfrac { 4 }{ 5 } $$

$$ \dfrac { r+2 }{ n-r-1 } =\dfrac { 4 }{ 5 } $$

$$ On\quad solving\quad above\quad we\quad get\quad r=26,n=62$$

$$ 2n+3r=238$$$$ $$
Question 3
1 / -0

The number of permutations of the letters of the word $$HONOLULU$$ taken $$4$$ at a time is

A
$$ 354$$

B
$$ 314$$

C
$$ 210$$

D
$$ 124$$

Solution

$$H-1$$
$$N-1$$
$$O-2$$
$$L-2$$
$$U-2$$
Number of permutation $$=\dfrac {n!}{r_1\times r_2\times....\times r_n} =\dfrac { 8! }{ { (2! })^{ 3 }4! } =210$$
Question 4
1 / -0

The total number of different combinations of one
or more letters which can be made from the letter of the word MISSISSIPPI is,

A
$$150$$

B
$$148$$

C
$$149$$

D
$$146$$

Solution

We have 1M, 4I ,4S , 2P
Therefore total number of selection of one or more letters=(1+1)(4+1)(4+1)(2+1)-1=149
Question 5
1 / -0

Find the remainder when $$105!$$ is divided by $$214.$$

A
$$168$$

B
$$108$$

C
$$196$$

D
$$172$$
Question 6
1 / -0

We wish to select $$6$$ persons from $$8$$, but if the person A is chosen, then B must be chosen. In how many ways can selections be made?

A
$$20$$

B
$$22$$

C
$$24$$

D
$$26$$

Solution

When A is not Taken then B may be Chosen (No of remaining Candidates $$=8-1=7$$)
No.of selections $$={}^7C_6=7$$

When A is taken B must be chosen (No of remaining Candidates $$=6$$ and number of remaining selections $$=4$$)
No. of selections $$={}^6C_4=15$$

Total no. of ways $$=15+7=22$$
Question 7
1 / -0

Number of arrangements of the letter $$HOLLYWOOD$$ in which all $$Os$$ are separated.

A
$$360\ ^{ 7 }{ C }_{ 3 }$$

B
$$15\ ^{ 7 }{ P }_{ 4 }$$

C
$$15\ ^{ 7 }{ C }_{ 4 }$$

D
$$30\ ^{ 7 }{ P }_{ 4 }$$
Question 8
1 / -0

$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\dfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\dfrac { { C }_{ 3 } }{ { C }_{ 2 } } +....+15\dfrac { { C }_{ 15 } }{ { C }_{ 14 } }$$ is equal to

A
$$100$$

B
$$120$$

C
$$-120$$

D
$$None\ of\ these$$

Solution

$$\begin{array}{l} \frac { { { C_{ 1 } } } }{ { { C_{ 0 } } } } +\frac { { 2{ C_{ 2 } } } }{ { { C_{ 1 } } } } +\frac { { B{ C_{ 3 } } } }{ { { C_{ 2 } } } } ........15\frac { { { C_{ 13 } } } }{ { { C_{ 14 } } } } \\ \frac { { { C_{ 1 } } } }{ { { C_{ 0 } } } } =4\, \\ \frac { { 2{ C_{ 2 } } } }{ { { C_{ 1 } } } } =n-1 \\ \frac { { 15{ C_{ 15 } } } }{ { { C_{ 14 } } } } =1 \\ n+\left( { n-1 } \right) ....1 \\ =\frac { { 15\left( { 15+1 } \right) } }{ 2 } \\ =120 \end{array}$$
Question 9
1 / -0

How many committee of five persons with a chairperson can be selected from $$12$$ persons.

A
$$924$$

B
$$825$$

C
$$736$$

D
$$643$$

Solution

No of Persons $$=5+1$$(Chairperson)

$$\therefore$$ No of committees that can be selected $$={}^{12}C_6=924$$
Question 10
1 / -0

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?

A
$$50000$$

B
$$50100$$

C
$$50300$$

D
$$50400$$

Solution

$$2$$ vowels can be chosen in $$5{c}_{2}$$ ways.
$$2$$ consonants can be chosen in $$21{c}_{2}$$ ways.
$$4$$ letter can be arranged in $$4$$ ways.
$$\therefore$$ The no of words containing $$2$$ vowels and $$2$$ consonants $$=5{c}_{2}\times 21{c}_{2}\times 4!=10\times 210\times 24=50400$$

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Permutations and Combinations Test 25

Result

Permutations and Combinations Test 25

Score

-

Rank

-

SHARING IS CARING

Question 1

$$c^{2}=a(b+c)$$

$$a^{2}=c(a+b)$$

$$b^{2}=c(a-b)$$

$$\dfrac {1}{a}+\dfrac {1}{b}+\dfrac {1}{c}=1$$

Solution

Question 2

$$238$$

$$220$$

$$203$$

$$240$$

Solution

Question 3

$$ 354$$

$$ 314$$

$$ 210$$

$$ 124$$

Solution

Question 4

$$150$$

$$148$$

$$149$$

$$146$$

Solution

Question 5

$$168$$

$$108$$

$$196$$

$$172$$

Question 6

$$20$$

$$22$$

$$24$$

$$26$$

Solution

Question 7

$$360\ ^{ 7 }{ C }_{ 3 }$$

$$15\ ^{ 7 }{ P }_{ 4 }$$

$$15\ ^{ 7 }{ C }_{ 4 }$$

$$30\ ^{ 7 }{ P }_{ 4 }$$

Question 8

$$100$$

$$120$$

$$-120$$

$$None\ of\ these$$

Solution

Question 9

$$924$$

$$825$$

$$736$$

$$643$$

Solution

Question 10

$$50000$$

$$50100$$

$$50300$$

$$50400$$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.