Permutations and Combinations Test 39

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Permutations and Combinations Test 39

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Weekly Quiz Competition

Question 1
1 / -0

$$12$$ persons meet in a room and each shakes hands with all the others. How many handshakes are there?

A
$$144$$

B
$$132$$

C
$$72$$

D
$$66$$

Solution

Number of handshakes $$={^{12}C_2}=\dfrac{12\times 11}{2}=66$$.
Question 2
1 / -0

Out of $$5$$ men and $$2$$ women, a committee of $$3$$ is to be formed. In how many ways can it be formed if at least one woman is included in each committee?

A
$$21$$

B
$$25$$

C
$$32$$

D
$$50$$

Solution

We may have:
(i) $$1$$ woman and $$2$$ men or (ii) $$2$$ women and $$1$$ man.
$$\therefore$$ required number of ways$$=(^2C_1\times {^5C_2})+(^2C_2\times {^5C_1})=\left(2\times \dfrac{5\times 4}{2\times 1}\right)+(1\times 5)$$
$$=(20+5)=25$$.
Question 3
1 / -0

If a denotes the number of permutation of $$x+2$$ things taken all at a time, $$b$$ the number of permutation of x things taken 11 at a time and c the number of permutation of $$x-11$$ things taken all at a time such that $$a=182 bc$$, then the value of $$x$$ is

A
$$15$$

B
$$12$$

C
$$10$$

D
$$18$$

Solution
Question 4
1 / -0

The exponent of 3 in $$100!$$ is

A
$$12$$

B
$$24$$

C
$$48$$

D
$$96$$

Solution
Question 5
1 / -0

The total number of 5- digit telephone numbers that can be composed with distinct digits, is

A
$$ ^{10}P_{2}$$

B
$$ ^{10}P_{5}$$

C
$$ ^{10}C_{2}$$

D
None of these

Solution

Question 6

1 / -0

The letters of word 'ZENITH' are written in all positive ways. If all these words are written in the order of a dictionary, then the rank of the word 'ZENITH' is

$$716$$

$$692$$

$$698$$

$$616$$

Solution

The total number of words is $$6! = 720$$. Let us write the letters of word ZENITH alphabetically, i.e, EHINTZ.

For ZENITH word starts with	Word starting with	Number of words
$$Z$$	$$E$$	$$5!$$
	$$H$$	$$5!$$
	$$I$$	$$5!$$
	$$N$$	$$5!$$
	$$T$$	$$5!$$
ZEN	ZEH	$$3!$$
	ZEI	$$3!$$
ZENI	ZENH	$$2$$
ZENIT	ZENIH	$$1$$
	Total number of words before ZERNITH	$$615$$

Hence, there are $$615$$ words before ZENITH, so the rank of ZENITH is $$616$$.

Question 7
1 / -0

If $$^{n}C_{3} + ^{n}C_{4} > ^{n + 1}C_{3}$$, then

A
$$n > 6$$

B
$$n > 7$$

C
$$n < 6$$

D
None of these

Solution

$$^{n}C_{3} + ^{n}C_{4} > ^{n + 1}C_{3}$$
$$\Rightarrow ^{n + 1}C_{4} > ^{n + 1}C_{3} (\because ^{n}C_{r} + ^{n}C_{r + 1} = ^{n + 1}C_{r + 1})$$
$$\Rightarrow \dfrac {^{n + 1}C_{4}}{^{n + 1}C_{3}} > 1$$
$$\Rightarrow \dfrac {n - 2}{4} > 1$$
$$\Rightarrow n > 6$$.
Question 8
1 / -0

Find the values of $$ ^{61}C_{57} -^{60}C_{56} $$

A
$$ ^{61}C_{58} $$

B
$$ ^{60}C_{57} $$

C
$$ ^{60}C^{58} $$

D
$$ ^{60}C_{56} $$

Solution

$$ ^{61}C_{57} -^{60}C_{56} = ^{60+1}C_{57} -^{60}C_{56} $$
$$ =(^{60}C_{57} + ^{60}C_{56}) - ^{60}C_{56} $$
$$ ( \because ^{n+1}C_r = ^nC_r + ^nC_{r-1}) $$
$$ =^{60}C_{57} + ^{60}C_{56} -^{60}C_{56} $$
$$ =^{60}C_{57} $$
hence option (B) correct.
Question 9
1 / -0

If $$ ^{15}C_{3r} = ^{15}C_{r+3} $$ then r equal to :

A
$$ 5 $$

B
$$ 4 $$

C
$$ 3 $$

D
$$ 2 $$

Solution

$$ ^{15}C_{3r} =^{15}C_{r+3} $$
$$ \Rightarrow ^{15}C_{3r} =^{15}C_{15-(r+3) } ( \because ^nC_r = ^nC_{n-r}) $$
$$ \Rightarrow ^{15}C_{3r} =^{15}C_{12-r} $$
On comparing $$ 3r = 12 -r $$
$$ \Rightarrow 3r +r = 12 $$
$$ \Rightarrow 4r = 12 $$
$$ \Rightarrow r = 3 $$
hence option (C) is correct.
Question 10
1 / -0

$$ ^{ 47 }C_{ 4 }$$ +$$ \sum _{ r=1 }^{ 5 }$$ .$$^{ 52-r }C_{ 3 } $$is equal to :

A
$$ ^{51}C_4 $$

B
$$ ^{52}C_4 $$

C
$$ ^{53}C_4 $$

D
None of these

Solution

$$ = ^{47}C_4+ ^{ 52-1}C_3 + ^{52-2}C_3 + ^{52-3}C_3 + ^{52-4}C_3 +^{52-5}C_3 $$
$$ =^{47}C_4 + ^{51}C_3 + ^{50}C_3 + ^{49}C_3 +^{4}C_3 + ^{47}C_3 $$
$$ =(^{47}C_4 + ^{47}C_3) + ^{4}C_3 +^{49}C_3 +^{50}C_3 +^{51}C_3 $$
$$ = (^{48}C_4 +^{48}C_3 + ^{49}C_3 +^{50}C_3 +^{51}C_3 $$
$$( \because ^nC_r + ^nC_{r-1} = ^{n+1}C_r) $$
$$ = (^{49}C_4 +^{49}C_3) + ^{50}C_3 + ^{51}C_3)( \because ^nC_r + ^nC_{r-1} = ^{n+1}C_r) $$
$$ =(^{50}C_4 + ^{50}C_3) + ^{51}C_3 )(\because ^nC_r +^nC_{r-1} = ^{n+1}C_r) $$
$$ =(^{51}C_4 + ^{51}C_3 \quad ( \because ^nC_r +^nC_{r-1} = ^{n+1}C_r )$$
$$ =^{52}C_4 \quad ( \because ^nC_r + ^nC_{r-1} = ^{n+1}C_r )$$

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Re-Attempt Weekly Quiz Competition

Permutations and Combinations Test 39

Result

Permutations and Combinations Test 39

Score

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SHARING IS CARING

Question 1

$$144$$

$$132$$

$$72$$

$$66$$

Solution

Question 2

$$21$$

$$25$$

$$32$$

$$50$$

Solution

Question 3

$$15$$

$$12$$

$$10$$

$$18$$

Solution

Question 4

$$12$$

$$24$$

$$48$$

$$96$$

Solution

Question 5

$$ ^{10}P_{2}$$

$$ ^{10}P_{5}$$

$$ ^{10}C_{2}$$

None of these

Solution

Question 6

$$716$$

$$692$$

$$698$$

$$616$$

Solution

Question 7

$$n > 6$$

$$n > 7$$

$$n < 6$$

None of these

Solution

Question 8

$$ ^{61}C_{58} $$

$$ ^{60}C_{57} $$

$$ ^{60}C^{58} $$

$$ ^{60}C_{56} $$

Solution

Question 9

$$ 5 $$

$$ 4 $$

$$ 3 $$

$$ 2 $$

Solution

Question 10

$$ ^{51}C_4 $$

$$ ^{52}C_4 $$

$$ ^{53}C_4 $$

None of these

Solution

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