Interview Questions.

Cgi Group Aptitude Placement Papers - Cgi Group Aptitude Interview Questions and Answers


Cgi Group Aptitude Placement Papers - Cgi Group Aptitude Interview Questions and Answers

Q1. In A Box, There Are 5 Black Pens, 3 White Pens And 4 Red Pens. In How Many Ways Can 2 Black Pens, 2 White Pens And 2 Red Pens Can Be Chosen?

Number of methods of choosing 2 black pens from five black pens in 5C<sub>2</sub>approaches.

Number of approaches of choosing 2 white pens from 3 white pens in 3C<sub>2</sub>ways.

Number of methods of choosing 2 crimson pens from four pink pens in 4C<sub>2</sub>methods.

By the Counting Principle, 2 black pens, 2 white pens, and a pair of pink pens may be chosen in 10 x 3 x 6 =180 ways.

Q2. Ramesh Bought 10 Cycles For Rs. 500 Each. He Spent Rs. 2,000 On The Repair Of All Cycles. He Sold Five Of Them For Rs. 750 Each And The Remaining For Rs. 550 Each. Then The Total Gain Or Loss % Is?

Total CP = Rs. (500 X 10 + 2000) = Rs. 7000

SP = Rs. (five X 750 + five X 550) = Rs. 6500

Loss = CP - SP = 7000 - 6500 = 500

Loss Percent = 500/7000 X a hundred = 50/7.

Q3. There Are 5 Novels And four Biographies. In How Many Ways Can 4 Novels And 2 Biographies Can Be Arranged On A Shelf ?

Four novels can be decided on out of 5 in 5C4 methods.

2 biographies can be decided on out of four in 4C2 approaches.

Number of approaches of arranging novels and biographies = 5C4*4C2= 30 

After selecting any 6 books (four novels and a couple of biographies) in one of the 30 approaches, they can be arranged at the shelf in 6! = 720 methods.

By the Counting Principle, the total range of preparations = 30 x 720 = 21600.

Q4. In How Many Ways Can 4 Girls And five Boys Be Arranged In A Row So That All The Four Girls Are Together ?

Let 4 women be one unit and now there are 6 devices in all.

They can be arranged in 6! Ways.

In every of those arrangements four women may be arranged in four! Ways. 

Total range of arrangements in which women are usually together = 6! X four!= 720 x 24 = 17280.

Q5. A Fruit Seller Buys Lemons At 2 For A Rupee And Sells Them At 5 For Three Rupees. His Gain Percent Is?

Given, Cost charge (C.P) of two lemons = Rs. 1

=>C.P of one lemon = Rs. Half = Rs. 0.50

Given, Selling rate (S.P) of 5 lemons = Rs. 3

=>S.P of 1 lemon = Rs. Three/5 = 0.60

Gain = S.P of1 lemon -C.P of one lemon

= 0.60 - 0.50

= zero.10

=>Gain = Rs. Zero.10

Gain % = (Gain / C.P of one lemon) * 100%

= (0.10 / 0.50)* 100%

= 20%

Thus, Gain % = 20%.

Q6. By Selling Sugar At Rs. Five.Fifty eight Per Kg. A Man Loses 7%. To Gain 7% It Must Be Sold At The Rate Of Rs?

Cost of sugar = Rs 5.58 / kg

His lost percent =7 %

= one hundred - 7

= ninety three.

Gain percentage

= a hundred+ 7

= 107.

So, Gain = CP * benefit / one hundred - loss

= five.58 * 107 / one hundred - 7

= five.Fifty eight * 107 / 93

= 597.06 / 93

= 6.Forty two .Per kg 

6.Forty two .Consistent with kg is to be bought to advantage 7 % .

Q7. Find The Value Of 'n' For Which The Nth Term Of Two Ap's:15,12,nine.... And -15,-13,-eleven...... Are Equal?

Given are the 2 AP'S:

15,12,nine.... In which a=15, d=-3.............(1) 

-15,-13,-eleven..... In which a'=-15 ,d'=2.....(2) 

now the usage of the nth term's components,we get

a+(n-1)d = a'+(n-1)d'

substituting the price obtained in eq. 1 and a pair of,

15+(n-1) x (-three) = -15+(n-1) x 2

=> 15 - 3n + three = -15 + 2n - 2

=> 12 - 3n = -17 + 2n

=> 12+17 = 2n+3n

=> 29=5n

=> n= 29/five.

Q8. Consider The Word Rotor. Whichever Way You Read It, From Left To Right Or From Right To Left, You Get The Same Word. Such A Word Is Known As Palindrome. Find The Maximum Possible Number Of 5-letter Pa

The first letter from the proper may be chosen in 26 approaches because there are 26 alphabets. 

Having chosen this, the second one letter can be chosen in 26 methods 

The first  letters can selected in 26 x 26 = 676 approaches 

Having chosen the first  letters, the third letter can be selected in 26 approaches. 

All the three letters may be selected in 676 x 26 =17576 ways. 

It means that the most viable number of 5 letter palindromes is 17576 due to the fact the fourth letter is similar to the second letter and the fifth letter is similar to the first letter.

Q9. Out Of 7 Consonants And four Vowels, How Many Words Of 3 Consonants And 2 Vowels Can Be Formed?

Number of methods of choosing (3 consonants out of seven) and (2 vowels out of four) = (7C3*4C2) 

= 210. 

Number of agencies, every having three consonants and a couple of vowels = 210. 

Each institution carries 5 letters. 

Number of methods of arranging five letters amongst themselves = five! = a hundred and twenty 

Required wide variety of approaches = (210 x one hundred twenty) = 25200.

Q10. A College Has 10 Basketball Players. A 5-member Team And A Captain Will Be Selected Out Of These 10 Players. How Many Different Selections Can Be Made?

A crew of 6 members must be selected from the 10 players.

This can be finished in 10C6 or 210 ways. 

Now, the captain can be selected from those 6 players in 6 approaches.

Therefore, general ways the selection may be made is 210×6= 1260.

Q11. Find The Selling Price Of A Scooter Toy Which Is Bought For Rs. 1200 And Sold At Profit Of 30 %?

Given, Cost fee = Rs. 1200

Profit = 30%

Selling charge = [100 + gain%] / a hundred * Cost fee

= [130 / 100] * 1200

= one hundred thirty * 12

= 1560

Therefore, Selling charge = 1560.

Q12. How Many 7 Digit Numbers Can Be Formed Using The Digits 1, 2, 0, 2, 4, 2, four?

There are 7 digits 1, 2, zero, 2, 4, 2, 4 in which 2 happens 3 times, four takes place 2 instances.

Number of 7 digit numbers = 7!3!×2! = 420 

But out of these 420 numbers, there are a few numbers which begin with '0' and they may be not 7-digit numbers.

The wide variety of such numbers beginning with '0'. 

=6!Three!×2! = 60

Hence the desired quantity of 7 digits numbers = 420 - 60 = 360.

Q13. A Tradesman's Prices Are 20% Above C.P. He Allows His Customers Some Discount On His Bill And Makes A Profit Of eight%. The Rate Of Discount Is?

Let the Cost Price(C.P) be Rs. 100

Given,tradesman's prices are 20% above C.P

=> Marked Price (M.P) = 20% greater than C.P

=> M.P = Rs. A hundred and twenty

Given,income = eight%

=> Selling charge (S.P) = 8% greater than C.P

=> S.P = Rs. 108

Rate of Discount = (M.P - S.P) / M.P * a hundred%

= (one hundred twenty - 108) / 120 * a hundred%

= (12 / one hundred twenty) * one hundred%

= 10 %

Thus,Rate of Discount = 10%.

Q14. 5 Men And four Women Are To Be Seated In A Row So That The Women Occupy The Even Places . How Many Such Arrangements Are Possible?

There are total 9 locations out of which 4 are even and relaxation 5 places are unusual.

4 women may be organized at 4 even locations in four! Approaches.

And five guys may be placed in ultimate five locations in 5! Approaches.

Hence, the specified wide variety of diversifications  = 4! X 5! = 24 x one hundred twenty = 2880.

Q15. How Many Alphabets Need To Be There In A Language If One Were To Make 1 Million Distinct three Digit Initials Using The Alphabets Of The Language?

1 million awesome three digit initials are wished. 

Let the quantity of required alphabets in the language be ‘n’. 

Therefore, the use of ‘n’ alphabets we will shape n * n * n = n3 distinct 3 digit initials.

Note wonderful initials is different from initials wherein the digits are unique. 

For example, AAA and BBB are proper combos within the case of wonderful initials while they are not authorised whilst the digits of the initials want to be different. 

This n3 different initials = 1 million 

i.E. N3=106  (1 million = 106)

  => n = 102 = 100

Hence, the language wishes to have not less than one hundred alphabets to reap the goal.

Q16. How Many 4-letter Words With Or Without Meaning, Can Be Formed Out Of The Letters Of The Word, 'logarithms', If Repetition Of Letters Is Not Allowed?

'LOGARITHMS' consists of 10 specific letters.

Required variety of phrases = Number of arrangements of 10 letters, taking four at a time.

= 10P4

= 5040.

Q17. Find The Number Of Subsets Of The Set 1,2,three,4,five,6,7,8,nine,10,eleven Having 4 Elements?

Here the order of choosing the factors doesn’t depend and this is a problem in combos.

We have to discover the wide variety of approaches of selecting 4 elements of this set which has 11 factors.

This may be carried out in 11C4ways = 330 ways.

Q18. A Man Sold An Item For Rs. 1500 At A Loss Of 25%. What Will Be The Selling Price Of Same Item If He Sells It At A Profit Of 20 %?

S.P = seventy five % of CP

=> seventy five x CP /a hundred= 1500

=> CP = 2000

20 % of CP = (20/100) x 2000 = four hundred

SP = 2000 + four hundred = 2400.

Q19. The Profit Earned By Selling An Article For Rs. 600 Is Equal To The Loss Incurred When The Same Article Is Sold For Rs. 4

Let the value charge be Rs. Ok

Now, as in keeping with the question,

six hundred - okay = ok - 400

=> 2k = one thousand

=> okay = 500

Again, promoting price of the item for making 25 % earnings

= (500 x a hundred twenty five) /200

= 125 * five

= Rs. 625.

Q20. Toffee Are Bought At A Rate Of eight For One Rupee. To Gain 60% They Must Be Sold At?

Given, Cost rate (C.P) of eight toffees = Re. 1

Gain = 60%

So, Selling charge, (S.P) = [100 + Gain%] / a hundred * C.P

= Rs. (one hundred sixty / 100) x 1

= Rs. Eight / 5

For Rs. 8 / 5, toffees sold = eight

For Re. 1, toffees sold = (eight x 5) / eight = five

So, to benefit 60%, toffees must be bought at five for Re. 1.

Q21. A Team Of 8 Students Goes On An Excursion, In Two Cars, Of Which One Can Seat five And The Other Only

There are eight college students and the most capability of the cars collectively is nine.

We may divide the 8 college students as follows

Case I: five college students within the first vehicle and 3 inside the 2d 

Case II: four college students within the first car and 4 inside the 2nd

Hence, in Case I: eight college students are divided into agencies of five and 3 in8C3 approaches.

Similarly, in Case II: 8 college students are divided into two agencies of four and 4 in 8C4ways.


the entire wide variety of ways in which eight college students can travel is:

8C3+8C4=fifty six+70= 126.

Q22. A, B And C Entered Into Partnership In Business A Got three/five Of The Profit And B And C Distributed The Remaining Profit Equally. If C Got Rs. Four hundred Less Than A, The Total Profit Was?

Let x be the whole earnings

A’s proportion in income = Rs. 3x/5

Remaining Profit = x - (3x/5) = 2x/5

So, B’s percentage in profit = Rs. X/five

C’s proportion in income = Rs. X/five

Given,(3x/five – x/five) = 400

=> 2x/five = four hundred

=> x = (four hundred×5)/ 2

=> x = Rs. A thousand

Therefore, Total Profit = x =Rs. One thousand.

Q23. If The Letters Of The Word Chasm Are Rearranged To Form 5 Letter Words Such That None Of The Word Repeat And The Results Arranged In Ascending Order As In A Dictionary What Is The Rank Of The Word Cha

The five letter phrase may be rearranged in 5!=a hundred and twenty Ways with none of the letters repeating.

The first 24 of those phrases will start with A.

Then the 25th phrase will start will CA _ _ _. 

The closing three letters may be rearranged in three!=6 Ways. I.E. 6 phrases exist that start with CA.

The next phrase starts with CH after which A, i.E., CHA _ _. 

The first of the words can be CHAMS. The subsequent phrase will be CHASM.

Therefore, the rank of CHASM may be 24+6+2= 32.

Q24. There Are 2 Brothers Among A Group Of 20 Persons. In How Many Ways Can The Group Be Arranged Around A Circle So That There Is Exactly One Person Between The Two Brothers?

Restoration one man or woman and the brothers B1 P B2 = 2 ways to accomplish that. 

Different 17 human beings= 17! 

Each man or woman out of 18 may be fixed between the two=18, therefore, 2 x 17! X 18=2 x 18!.

Q25. From five Consonants And four Vowels, How Many Words Can Be Formed Using three Consonants And 2 Vowels ?

From five consonants, 3 consonants may be selected in 5C3 approaches.

From 4 vowels, 2 vowels may be decided on in 4C2ways.

Now with every choice, range of ways of arranging five letters is 5P5ways.

Total variety of words = 5C3*4C2*5P5= 10x 6 x five x 4 x three x 2 x 1= 7200.

Q26. The Indian Cricket Team Consists Of sixteen Players. It Includes 2 Wicket Keepers And five Bowlers. In How Many Ways Can A Cricket Eleven Be Selected If We Have To Select 1 Wicket Keeper And Atleast four Bowlers

We are to choose eleven gamers which includes 1 wicket keeper and 4 bowlers  or, 1 wicket keeper and five bowlers.

Number of approaches of choosing 1 wicket keeper, four bowlers and six different players in 2C1*5C4*9C6= 840

Number of ways of selecting 1 wicket keeper, five bowlers and 5 different players in 2C1*5C5*9C5=252

Total number of methods of choosing the group = 840 + 252 = 1092.

Q27. Suppose You Can Travel From A Place A To A Place B By 3 Buses, From Place B To Place C By 4 Buses, From Place C To Place D By 2 Buses And From Place D To Place E By 3 Buses. In How Many Ways Can You T

The bus fromA to B may be decided on in three ways.

The bus from B to C can be selected in 4 approaches.

The bus from C toD can be decided on in 2 ways.

The bus fromD to E can be selected in three methods.

So, by way of the General Counting Principle, it is easy to travel fromA to E in three x four x 2 x 3 methods = 72.

Q28. A Man Sold A Chair At A Loss Of 6%. Had He Been Able To Sell It At A Gain Of 10%, It Would Have Fetched Rs. 96 More Than It Did. What Was His Cost Price?

In the given hassle, permit C.P denote the price charge,

then (a hundred +10)% of CP -(a hundred-6) % of CP = Rs. 96

=>(a hundred and ten)% of CP - (94) % of CP = Rs.Ninety six

=>16 % of CP = ninety six

=> sixteen / one hundred = ninety six

=> CP = 96 x a hundred / sixteen

=> 9600 / sixteen

= six hundred Rs

Rs 600is fee charge.

Q29. Suppose You Want To Arrange Your English, Hindi, Mathematics, History, Geography And Science Books On A Shelf. In How Many Ways Can You Do It?

We have to arrange 6 books.

The range of diversifications of n items is n! = n. (n – 1) . (n – 2) ... 2.1

Here n = 6 and consequently, range of variations is 6.5.Four.Three.2.1 = 720.

Q30. A Man Sold A Horse At A Loss Of four%. Had He Been Able To Sell It At A Gain Of 12%, It Would Have Fetched Rs. Sixty four More Than It Did. What Was His Cost Price?

In the given problem,

Let C.P denote the fee fee,

Then (100+12)% of CP = (a hundred-4) % of Cost

=> Rs. 128 (112)% of CP = (96) % of Cost = Rs.128

16 % of CP = 128

=> CP = 128 x one hundred / sixteen

= 12800 / sixteen

= 800.

Q31. In How Many Ways Can The Letters Of The Word "hassle" Be Rearranged To Make 7 Letter Words Such That None Of The Letters Repeat?

There are seven positions to be stuffed.

The first function may be crammed the usage of any of the 7 letters contained in PROBLEM.

The second role can be stuffed with the aid of the last 6 letters as the letters should not repeat.

The 0.33 position may be stuffed by the remaining 5 letters simplest and so on.

Therefore, the overall variety of methods of rearranging the 7 letter phrase = 7*6*5*four*3*2*1 = 7! Ways.

Q32. How Many 6-digit Even Numbers Can Be Formed From The Digits 1, 2, 3, 4, 5, 6 And 7 So That The Digits Should Not Repeat And The Second Last Digit Is Even ?

Let ultimate digit is 2

while second ultimate digit is four last four digits may be filled in one hundred twenty approaches,

similarly 2nd remaining digit is 6 remained four digits may be crammed in 120 approaches.

So for remaining digit = 2, general numbers=240

Similarly for 4 and 6

When ultimate digit = 4, total no. Of approaches =240

and closing digit = 6, total no. Of approaches =240

so general of 720 even numbers are possible.

Q33. A Letter Lock Consists Of Three Rings Each Marked With Six Different Letters. The Number Of Distinct Unsuccessful Attempts To Open The Lock Is At The Most?

Since every ring consists of six exceptional letters,

the total quantity of tries viable with the three jewelry is = 6 x 6 x 6 = 2@Of these tries,

one in all them is a a hit try.

Maximum quantity of unsuccessful attempts = 216 - 1 = 215.

Q34. A Box Contains 2 White Balls, 3 Black Balls And four Red Balls. In How Many Ways Can three Balls Be Drawn From The Box, If At Least One Black Ball Is To Be Included In The Draw?

We can also have(1 black and 2 non-black) or (2 black and 1 non-black) or (three black).

Required variety of approaches=(3C1*6C2)+(3C2*6C1)+3C3 = (forty five + 18 + 1) =64.

Q35. How Many Different Four Letter Words Can Be Formed (the Words Need Not Be Meaningful Using The Letters Of The Word "mediterranean" Such That The First Letter Is E And The Last Letter Is R?

The first letter is E and the closing one is R.

Therefore, one has to locate  more letters from the closing 11 letters.

Of the 11 letters, there are 2 Ns, 2Es and 2As and one every of the remaining 5 letters.

The 2nd and third positions can both have two different letters or have each the letters to be the identical.

Case 1: When the two letters are extraordinary. One has to select two distinctive letters from the eight available exceptional choices. This may be done in 8 * 7 = 56 ways.

Case 2: When the two letters are identical. There are 3 alternatives - the three can be either Ns or Es or As. Therefore, 3 approaches.

Total quantity of opportunities = 56 + 3 = 59.

Q36. When Four Fair Dice Are Rolled Simultaneously, In How Many Outcomes Will At Least One Of The Dice Show three?

When four dice are rolled simultaneously, there may be a complete of 6 x 6 x 6 x 6 = 1296 outcomes.

The quantity of outcomes in which not one of the 4 cube display 3 will be five x 5 x 5 x five = 625 outcomes.

Therefore, the range of outcomes wherein as a minimum one die will display 3 = 1296 – 625 = 671.

Q37. If A Shirt Costs Rs. Sixty four After 20% Discount Is Allowed, What Was Its Original Price In Rs?

Let the fee be Rs a hundred

After cut price of 20% we get = 100-20 = eighty

blouse prices Rs. Sixty four

Let x be the fee fee of the blouse,

x * 80/one hundred = sixty four

x = (64 x a hundred) / 80 = 80

Original charge of shirt in Rs. 80.

Q38. If There Is A Profit Of 20% On The Cost Price Of An Article, The Percentage Of Profit Calculated On Its Selling Price Will Be?

Let the Cost Price be Rs. 100,

Sincethere is a income of 20% at the Cost Price,

then Selling Price = C.P + 20% of C.P

= a hundred + 20

= Rs. 120

=>Selling Price =Rs. 120

Gain = SP - CP 

= a hundred and twenty - one hundred

= 20

Gain % on S.P = (Gain / S.P) * a hundred%

= (20/120) x 100

= 50/three%.

Q39. A Committee Of five Persons Is To Be Formed From 6 Men And four Women. In How Many Ways Can This Be Done When At Least 2 Women Are Included ?

When at the least 2 ladies are blanketed.

The committee may additionally consist of three girls, 2 guys : It can be done in  4C3*6C2ways

or, four girls, 1 guy : It may be achieved in  4C4*6C1ways

or, 2 women, 3 men : It can be done in 4C2*6C3ways.

Total range of ways of forming the committees

= 4C2*6C3+4C3*6C2+4C4*6C1

= 6 x 20 + four x 15 + 1x 6

= one hundred twenty + 60 + 6 =186.

Q40. From A Total Of Six Men And Four Ladies A Committee Of Three Is To Be Formed. If Mrs. X Is Not Willing To Join The Committee In Which Mr. Y Is A Member, Whereas Mr.Y Is Willing To Join The Committee O

We first count the variety of committee wherein

(i). Mr. Y is a member 

(ii). The ones wherein he is not

Case (i): As Mr. Y consents to be in committee most effective in which Mrs. Z is a member.

Now we are left with (6-1) guys and (4-2) girls (Mrs. X isn't always willing to join).

We can choose 1 greater in5+2C1=7 ways.

Case (ii): If Mr. Y isn't a member then we left with (6+four-1) human beings. 

We will choose 3 from nine in 9C3=eighty four approaches.

Thus, overall wide variety of methods is 7+84= ninety one approaches.

Q41. In How Many Ways Can five Letters Be Posted In four Letter Boxes?

First letter can be published in 4 letter packing containers in 4 ways.

Similarly 2nd letter may be posted in 4 letter bins in 4 methods and so on.

Hence all of the five letters may be published in = four x four x 4 x 4 x 4 = 1024.

Q42. Find The Total Number Of Distinct Vehicle Numbers That Can Be Formed Using Two Letters Followed By Two Numbers. Letters Need To Be Distinct?

Out of 26 alphabets  distinct letters may be selected in 26P2 methods.

Coming to numbers part, there are 10 ways.

(any range from zero to 9 may be chosen) to pick out the first digit and further any other 10ways to pick the second one digit.

Hence there are completely 10X10 = a hundred ways. 

Combined with letters there are 6P2 X 100 methods = 65000 approaches to select car numbers.

Q43. There Are Three Rooms In A Hotel: One Single, One Double And One For Four Persons. How Many Ways Are There To House Seven Persons In These Rooms?

Choose 1 person for the unmarried room & from the remaining select 2 for the double room & from the ultimate select four people for the 4 man or woman room,

Then, 7C1 x 6C2 x 4C4 

= 7 x 15 x 1 = one zero five.

Q44. Compute The Sum Of 4 Digit Numbers Which Can Be Formed With The Four Digits 1,3,5,7, If Each Digit Is Used Only Once In Each Arrangement?

The quantity of preparations of four specific digits taken four at a time is given by means of 4P4 = four! = 24.

All the four digits will arise identical number of times at every of the position,particularly ones,tens,loads,thousands.

Thus,every digit will occur 24/4 = 6 instances in each of the position.

The sum of digits in a single's position will be 6 x (1+3+five+7) = 96.

Similar is the case in ten's,hundred's and thousand's locations.

Therefore,the sum may be ninety six + 96 x 10 + ninety six x one hundred + ninety six x 100 = 106656.

Q45. An Article Was Sold At A Loss Of five%.If It Were Sold For Rs. 30 More ,the Gain Would Have Been 1.25%. The Cost Price Of The Article Is?

Let, fee price of the article = Rs.100x

Then selling fee = five% lack of Cost price

= C.P - loss

= 100x - (5/a hundred)*100x

= 100x - 5x

= 95x

=> promoting rate = 95x

But if he offered the product for Rs.30 more, ==> his income is 1.25%.

In this example his promoting rate = 100x + (1.25/a hundred) * 100x

= 100x + 1.25x

= one hundred and one.25x

=> promoting charge = one zero one.25x

Difference in  selling costs = Rs.30

=> one hundred and one.25x - 95x = Rs.30

=> 6.25x = Rs.30

=> x = Rs.30 / 6.25

=> x = Rs.4.8 --->Substitutingin cost rate, we get

Cost Price of the thing = Rs. 100x = Rs. A hundred * 4.Eight = Rs. 480

Therefore, Cost Price of the thing = Rs. 480.

Q46. How Many Arrangements Can Be Made Out Of The Letters Of The Word Committee, Taken All At A Time, Such That The Four Vowels Do Not Come Together?

There are total 9 letters within the phrase COMMITTEE wherein there are 2M's, 2T's, 2E's.

The quantity of approaches wherein nine letters may be arranged = nine!2!×2!×2! = 45360

There are 4 vowels O,I,E,E in the given word. If the 4 vowels constantly come collectively, taking them as one letter we should arrange 5 + 1 = 6 letters which include 2Ms and 2Ts and this be completed in 6!2!×2! = a hundred and eighty ways.

In which of a hundred and eighty approaches, the 4 vowels O,I,E,E remaining collectively can be arranged in four!2! = 12 ways.

The quantity of approaches wherein the four vowels continually come collectively = a hundred and eighty x 12 = 2160.

Hence, the desired range of methods wherein the 4 vowels do not come together = 45360 - 2160 = 43200.

Q47. If Repetition Of The Digits Is Allowed, Then The Number Of Even Natural Numbers Having Three Digits Is?

In a three digit number one’s region may be crammed in 5 exceptional ways with (0,2,4,6,eight)

10’s area may be stuffed in 10 exclusive methods

100’s vicinity can be stuffed in nine exclusive methods

There fore overall variety of approaches = 5X10X9 = 450.