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Mettl Permutation and Combination Quiz 1

Question 1

Time: 00:00:00

There are 4 spades, 7 hearts, and 6 diamonds in a set of cards. In how many ways can a person make a selection of cards from the set.

269

280

279

281

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Solution:
Zero or more spades can be selected in 4+ 1 = 5 ways (0 spade, 1 spade, 2 spades, 3 spades, 4 spades)
Similarly,
Hearts can be selected in 7+1 = 8 ways
Diamonds in 6 + 1 = 7 ways
So, total no. of ways = 5*7*8 = 280
But in case we included 0 spades, 0 heart, 0 diamond, so we have to subtract that case
Hence, 280 - 1 = 279 ways

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Question 2

Time: 00:00:00

In How many ways 4 Maruti, 5 BMW, and 7 Audi’s can be arranged in a row so that cards of the same company are arranged together?

4! 5! 7! 3!

4! 5! 7!

4! 6! 7! 3!

None of these

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Solution:
4 Maruti’s can be arranged in 4! Ways, similarly 5 BMW’s in 5! ways, 7 Audi’s in 7! ways, and now all these 3 companies can be arranged in 3! ways.
Hence, the correct answer will be, 4! 5! 7! 3!

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Question 3

Time: 00:00:00

In how many ways 5 Cycles and 5 Scooter are placed along a circular platform, so they are placed in alternative positions

5! 5!

6! 4!

5! 4! 2!

4! 5!

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Solution:
First Cycle can be placed along with the circular platform in 4! ways and now 5 Scooters can be placed in 5! ways. So, 4! 5! ways.

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Question 4

Time: 00:00:00

When Issac Newton goes for Paragliding he got fascinated by the term and thought in how many ways can the letters of the word ‘PARAGLIDING’ be arranged such that all the vowels occur together?

88322 ways

120960 ways

740 ways

144868

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Solution:
In the word “PARAGLIDING” there are 11 letters in which there are 4 vowels (i.e. 2 A’s and 2 I’s) and 7 consonants (i.e 2 G’s and each of P, R, L, D, N)
Considering vowel as one letter, the number of letters becomes 8 which can be arranged as
$\frac{8!}{2!} = \frac{40320}{2} = 20160$
Vowel A and I appear twice , so vowels can be arranged as
\frac{4!}{2! \times 2!} = \frac{24}{4} = 6
Hence the required number of ways in which the letters of the word “PARAGLIDING” be arranged so that all the vowels occur together = 20160 × 6 = 120960

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Question 5

Time: 00:00:00

Vishwanath Anand instead of playing Chess today thought of playing Snake & Ladder game; he is on 91 and has to get to 100 to complete the game. There is a snake on 93 and 96. In how many ways he can complete the game, if he doesn’t want to roll the dice more than three times.

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Solution:

91 --92 — 93 — 94 — 95 — 96 — 97 — 98 — 99 — 100

Total position advance needed = 100 – 91 = 9

One roll of dice can’t complete the game.

If he completes two rolls of dice.

Possible dice throws are – (3&6), (4&5), (5&4), (6&3)

But (5&4) will bring the token on 96, so this is rejected.

If he completes the game in three roll of dices

First dice reading options are 1,3,4,6

After checking all option and rejecting those in which token reaches on 93 or 96

Possible dice throws are (1,2,6), (1,3,5),(1,5,3), (1,6,2) ;

(3,1,5),(3,3,3),(3,4,2),(3,5,1);

(4,2,3),(4,3,2),(4,4,1)

(6, 1, 2), (6, 2, 1)

Total number of ways = 16

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Question 6

Time: 00:00:00

Viswanathan Anand in World Chess Championship notices, a chess board has rows and columns marked A to H and 1-8. Aman has a knight and a rook which he has to place on the board such that the two pieces are not in the same row or column, what is the total number of ways he can place the two pieces?

3072

3136

6272

6144

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Solution:
As shown in the image a knight and a rook has to be placed, but not in the same row or column.

Let us select any box out of 64 for placing knight, no of ways = $^{64}C_{1}$
Now, row 6 and column c can’t be used to place rooks. Remaining boxes = 64 - (8 + 7) = 49

The rook can be place in any of 49 boxes, no of ways = $^{49}C_{1}$
Total number of possible ways = $^{49}C_{1} \times ^{64}C_{1} = 3136$

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Question 7

Time: 00:00:00

In Indian Institute of Technology Main exam 2021 with all the questions mandatory to be answered can be attempted in 127 ways such that the student gets at least one question right. Find the number of ways in which he can answer 4 questions correctly.

Can't be determined

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Solution:
Any question can be answered in 2 ways (right or wrong)

Let the number of questions be N

$2^{N} - 1 = 127$
Therefore N = 7

Number of ways in answering 4 answers correctly = $^{7}C_{4} = 35$

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Question 8

Time: 00:00:00

A postmaster wants to get delivered 6 letters at six different addresses. In the post office there are 2 postmen then in how many ways can the postmaster send the letters at different addresses through the postmen?

$\frac{6!}{2!}$

$6! \times 2!/latex]</p></div><div class="checkmark"></div><div class="ls-progress-bar"><div class="progress-bar txtblack" role="progressbar" style="" aria-valuenow="25" aria-valuemin="0" aria-valuemax="100"></div> <div class="answer-title-progress"><p>6! \times 2!/latex]</p></div><div class="precentage"> </div> </div></div></div></div><div class="row ls-flex"><div class="col-lg-6 col-md-12 col-sm-12 col-xs-12 mq-answer-div answer-id-440776" data-ansid="440776" data-qstid="8"><div class="answer progress" id="quiz-ans-8-7"><input type="radio" name="ques_ans[7]" value="440776" class="radiobutton anslabel" id="qst8-7" name="answer_id" data-questionid="101367" data-questioncount="8" data-answerid="440776"><div class="ls-ans-title displaynone"><p>64</p></div><div class="checkmark"></div><div class="ls-progress-bar"><div class="progress-bar txtblack" role="progressbar" style="" aria-valuenow="25" aria-valuemin="0" aria-valuemax="100"></div> <div class="answer-title-progress"><p>64</p></div><div class="precentage"> </div> </div></div></div><div class="col-lg-6 col-md-12 col-sm-12 col-xs-12 mq-answer-div answer-id-440777" data-ansid="440777" data-qstid="8"><div class="answer progress" id="quiz-ans-8-8"><input type="radio" name="ques_ans[7]" value="440777" class="radiobutton anslabel" id="qst8-8" name="answer_id" data-questionid="101367" data-questioncount="8" data-answerid="440777"><div class="ls-ans-title displaynone"><p>36</p></div><div class="checkmark"></div><div class="ls-progress-bar"><div class="progress-bar txtblack" role="progressbar" style="" aria-valuenow="25" aria-valuemin="0" aria-valuemax="100"></div> <div class="answer-title-progress"><p>36</p></div><div class="precentage"> </div> </div></div></div></div></div></div></div></div><div class="help-links-div"><div class="ls-mt-correct-content"><p><img src="https://prepinsta.com/wp-content/plugins/lexcis-mtouch-quiz/images/happy-emoji.png"><span>Good Job!</span></p></div><div class="ls-mt-wrong-content"><div class="for-desktop ls-mtouch-wrong-banner"> <div class="card-body " style="padding:2px !important"> <div class="owl-carousel right-widget-banner line-dots"><div class="item"> <a href="https://prepinstaprime.com/purchase?utm_source=prepinsta&utm_medium=quiz+banner&utm_campaign=prime+traffic" target="_blank"> <img src="https://prepinstadotcom.s3.ap-south-1.amazonaws.com/wp-content/uploads/2023/08/Frame-985443-1.webp" /> </a> </div><div class="item"> <a href="https://prepinstaprime.com/purchase?utm_source=prepinsta&utm_medium=quiz+banner&utm_campaign=prime+traffic" target="_blank"> <img src="https://prepinstadotcom.s3.ap-south-1.amazonaws.com/wp-content/uploads/2023/08/Frame-985440-2.webp" /> </a> </div><div class="item"> <a href="https://prepinstaprime.com/purchase?utm_source=prepinsta&utm_medium=quiz+banner&utm_campaign=prime+traffic" target="_blank"> <img src="https://prepinstadotcom.s3.ap-south-1.amazonaws.com/wp-content/uploads/2023/08/Frame-985439-2.webp" /> </a> </div></div> </div> </div><div class="for-mobile ls-mtouch-wrong-banner"> <div class="card-body " style="padding:2px !important"> <div class="owl-carousel right-widget-banner line-dots"><div class="item"> <a href="https://prepinstaprime.com/purchase?utm_source=prepinsta&utm_medium=quiz+banner&utm_campaign=prime+traffic" target="_blank"> <img src="https://prepinsta.com/wp-content/uploads/2023/08/b.webp" /> </a> </div><div class="item"> <a href="https://prepinstaprime.com/purchase?utm_source=prepinsta&utm_medium=quiz+banner&utm_campaign=prime+traffic" target="_blank"> <img src="https://prepinsta.com/wp-content/uploads/2023/08/Complete-Placement-Prep-5.webp" /> </a> </div><div class="item"> <a href="https://prepinstaprime.com/purchase?utm_source=prepinsta&utm_medium=quiz+banner&utm_campaign=prime+traffic" target="_blank"> <img src="https://prepinsta.com/wp-content/uploads/2023/08/Complete-Placement-Prep-1-1.webp" /> </a> </div></div> </div> </div><p><img src="https://prepinsta.com/wp-content/plugins/lexcis-mtouch-quiz/images/sad-emoji.png"><span>Oops!</span></p></div><ul class="help-links-ul"><li><a href="https://prepinsta.com/permutation-combination/formulas/" class="help-link"><span>Formulas</span></a></li><li><a href="https://prepinsta.com/permutation-combination/tips-and-tricks-and-shortcuts/" class="help-link"><span>Shortcut, Tricks and Tips</span></a></li><li><a href="https://prepinsta.com/permutation-combination/how-to-solve-quickly" class="help-link"><span>How to solve quickly</span></a></li><li><a href="javascript:void(0)" class="help-link report reportquestionbtn" id="reportquestionbtn" data-qstid="101367" data-quizid="11893"><span>Report the answer</span></a></li></ul><ul class="help-links-ul"><li><a href="https://prepinsta.com/how-to-use-prepinsta-quiz-effectively/" class="help-link" target="_blank"><span>How to use PrepInsta Quiz</span></a></li></ul></div><div style="width: 100%; display: inline-block;"> <button class="btn pull-left anslabel skip_button" type="button" data-questioncount="8" data-questionid="101367" data-answerid="skipped" style=" padding: 6px 20px;width: auto;">Skip</button><button class="btn pull-right exp_hide_show" type="button" data-toggle="collapse" data-target="#explanation-8" aria-expanded="false" style="padding: 10px 25px;">Show Explanation <i class="fas fa-chevron-down" style="font-size: 10px; "></i></button></div><div class="explanation collapse in" id="explanation-8"><div class="toggle-tab tab text-center"><ul class="tabs quiz-tabs quiz-tabs101367 "><li class="tab-link current" data-tab="prep101367" data-mainid="101367"><div>PrepInsta Explanation</div></li><li class="tab-link" data-tab="user101367" data-mainid="101367"><div>User Explanation</div></li></ul></div> <div class="tab-content101367"><div id="prep101367" class="tab-content current tab-contentprep101367 prepexplanation_101367"><div class="default_prep101367">Once you attempt the question then PrepInsta explanation will be displayed.</div><div class="correct_prep101367" style="display:none"><p><strong>Solution:</strong><br /> Each letter can be delivered to the six different addresses in 2 different ways<br /> <br /> Hence, the required number of ways = 2^{6} = 64$

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Question 9

Time: 00:00:00

In Schrodinger's Cat Condition a box contains 2 white cats, 3 black cats and 4 red cats. In how many ways can 3 cats be drawn from the bag if at least one cat ball is to be included in the draw?

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Answer: From 2 white balls, 3 black balls, and 4 red balls, 3 balls are to be selected such that at least one black ball should be there.
Hence we have 3 choices as given below.
Choice 1: We can select 3 black balls.
Choice 2: We can select 2 black balls and 1 non-black ball.
Choice 3: We can select 1 black ball and 2 non-black balls.
Number of ways to select 3 black balls
$= ^{3}C_{3}$
Number of ways to select 2 black balls and 1 non-black ball
$= ^{3}C_{2} \times ^{6}C_{1}$
Number of ways to select 1 black ball and 2 non-black balls
$= ^{3}C_{1} \times ^{6}C_{2}$
Total number of ways
$= ^{3}C_{3} + ^{3}C_{2} \times ^{6}C_{1} + ^{3}C_{1} \times ^{6}C_{2}$
$= ^{3}C_{3} + ^{3}C_{1} \times ^{6}C_{1} + ^{3}C_{1} \times ^{6}C_{2} [∵ nCr = nC(n-r)]$
$= 1 + (3\times6) + [3 x (6\times5)/(2\times1)]$
$= 1 + 18 + 45$
$= 64$

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Question 10

Time: 00:00:00

Using all the letters of the word "DISCOUNT", how many different 8 letter words can be formed?

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Solution:
Total number of letters = 8
Using these letters the number of 8 letters words formed is = 8!

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