# Permutation Combination Questions and Answers updated daily – Aptitude

Permutation Combination Questions: Solved 27 Permutation Combination Questions and answers section with explanation for various online exam preparation, various interviews, Aptitude Category online test. Category Questions section with detailed description, explanation will help you to master the topic.

## Permutation Combination Questions

1. An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, …., 9 such that the first digit of the code is nonzero. The code, handwritten on a slip, can however potentially create confusion, when read upside down-for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?

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Correct Ans:71

Explanation:

The available digits are 0,1,2, …9.

The first digit can be chosen in 9 ways (0 not acceptable), the second digit can be accepted in 9 ways (digits repetition not allowed).

Thus, the code can be made in 9 × 9 = 81 ways.

Now there are only 4 digits 1, 6, 8, 9 which can create confusion.

Hence, the total number of codes which create confusion are = 4 × 3 = 12.

Out of these 12 codes 69 and 96 will not create confusion.

Hence, in total 12 – 2 = 10 codes will create confusion.

Hence, the total codes without confusion are 81 – 10 = 71.

The first digit can be chosen in 9 ways (0 not acceptable), the second digit can be accepted in 9 ways (digits repetition not allowed).

Thus, the code can be made in 9 × 9 = 81 ways.

Now there are only 4 digits 1, 6, 8, 9 which can create confusion.

Hence, the total number of codes which create confusion are = 4 × 3 = 12.

Out of these 12 codes 69 and 96 will not create confusion.

Hence, in total 12 – 2 = 10 codes will create confusion.

Hence, the total codes without confusion are 81 – 10 = 71.

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2. If a refrigerator contains 12 cans such that 7 blue cans and 5 red cans. In how many ways can we remove 8 cans so that at least 1 blue can and 1 red can remains in the refrigerator.

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Correct Ans:455

Explanation:

Given

Total No of Cans = 12

Total No of Red Cans = 7

Total No of Blue Cans = 5

**Given Conditi**on - No of Possible ways to draw 8 balls from the refrigerator which contains at least 1 blue and 1 red can

No of Possible ways are => (6,2) (5,3) (4,4)

=> 7 C 6 * 5 C2

=> 7 * 10

=> 70

=> 7C5 * 5 C3

=> 7 * 3 * 5 * 2

=> 21 * 10

=> 210

=> 7C4 * 5C4

=> 35 * 5

=> 175

Total Ways = 70 +210 + 175

=> 455.

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3. There is meeting of 20 delegates that is to be held in a hotel. In how many ways these delegates can be seated along a round table, if three particular delegates always seat together?

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Correct Ans:17! 3!

Explanation:

Give

Total 20 persons, 3 always seat together, 17 + 1 =18 delegates can be seated in (18 -1)! Ways = 17!

And now that three can be arranged in 3! Ways.

So, 17! 3! is the correct answer.

Total 20 persons, 3 always seat together, 17 + 1 =18 delegates can be seated in (18 -1)! Ways = 17!

And now that three can be arranged in 3! Ways.

So, 17! 3! is the correct answer.

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4. In how many ways a four digit even number can be formed by using the digits 2,3,5,8 exactly once?

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Correct Ans:12

Explanation:

**Solution is**

Given

Given

Four digit even number can be formed by using the digits 2,3,5,8

Since the number has to be an even number,the unit digit has to be 2 or 8.

First three places can be filled by remaining three digits.

= 3! + 3!

&nb

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5. In a plane there are totally 8 points (no three points are collinear), how many lines can be drawn ?

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Correct Ans:28

Explanation:

To draw a line we need 2 points.
From the given 8 points , we need to choose 2 points
Number of ways in which this can be done is 18C2= (8 x 7 ) / 2 = 28

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6. How many 4 digit even number can formed by using the digits 1,3,7 and 8 only once ?

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Correct Ans:6

Explanation:

We can form 24 numbers with digits 1,3,7 and 8 using only once.
To be an even number, the units digit has to even number, in this case only 8 should come in the unit's place . The remaining places, 3 positions can be filled in 3! , that is 6 ways

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7. How many words can be formed with or without meaning by taking all the letters from the word TAKEN ?

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Correct Ans:120

Explanation:

There are totally 4 letters in the word TAKEN
T, A, K, E and N.
No. Of Words = 5 x 4 x 3 x 2 x 1 = 120

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8. In how many different ways can the letters of the word 'OFFICES' be arranged ?

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Correct Ans:2520

Explanation:

The word OFFICES contains O,F, F I, C, E, S contains 7 letters, in which 2 are identical.
No. Of words that can formed = 7 ! / 2 ! = 2520

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9. Evaluate: 25! / 23!

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Correct Ans:600

Explanation:

25! / 23! = 23! X 24 x 25 / ( 23!) = 24 x 25 = 600

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10. In a party there were totally 20 people, each person shook his hands with the other person. How many hand shakes would have taken place ?

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Correct Ans:190

Explanation:

There are 20 people. Every person has to shake hands with the other person. Which means we have to find the number of ways of choosing 2 people from the 20.
The number of ways it ca happen = 20C2 = ( 20 x 19 ) / (1 x 2) = 190

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11. A letter lock consists of 4 rings, each ring contains 9 non-zero digits. This lock can be opened by setting a 4 digit code with the proper combination of each of the 4 rings Maximum how many codes can be formed to open the lock?

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Correct Ans:9 x 9 x 9 x 9

Explanation:

9 x 9 x 9 x 9 = 94

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12. How many words can be formed with or without meaning by taking all the letters from the word SMALL ?

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Correct Ans:60

Explanation:

Number of letters in SMALL is 5, but there are 2 L?s
Answer is 5! / 2! = 120 / 2 = 60

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13. In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

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Correct Ans:720

Explanation:

The word 'OPTICAL' contains 7 different letters.

When the vowels OIA are always together, they can be supposed to form one letter.

Then, we have to arrange the letters PTCL (OIA).

Now, 5 letters can be arranged in 5! = 120 ways.

The vowels (OIA) can be arranged among themselves in 3! = 6 ways.

Required number of ways

= (120 x 6)

= 720.

When the vowels OIA are always together, they can be supposed to form one letter.

Then, we have to arrange the letters PTCL (OIA).

Now, 5 letters can be arranged in 5! = 120 ways.

The vowels (OIA) can be arranged among themselves in 3! = 6 ways.

Required number of ways

= (120 x 6)

= 720.

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14. 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?

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Correct Ans:5040

Explanation:

LOGARITHMS' contains 10 different letters.

Required number of words = Number of arrangements of 10 letters, taking 4 at a time.

=

Required number of words = Number of arrangements of 10 letters, taking 4 at a time.

=

^{10}P_{4}= (10 x 9 x 8 x 7) = 5040.
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15. In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?

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Correct Ans:36

Explanation:

There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.
Let us mark these positions as under: (1) (2) (3) (4) (5) (6)
Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.
Number of ways of arranging the vowels = 3P3 = 3! = 6.
Also, the 3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = 3P3 = 3! = 6.
Total number of ways = (6 x 6) = 36.

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16. In how many ways can the letters of the word 'LEADER' be arranged ?

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Correct Ans:360

Explanation:

The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.

Therefore, Required number of ways = 6! / (1!)(2!)(1!)(1!)(1!) = 360

Therefore, Required number of ways = 6! / (1!)(2!)(1!)(1!)(1!) = 360

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17. How many numbers greater than a million can be formed by using the digits 7, 4, 6 and 0 if 4 has to be used twice, 6 has to be used thrice and the rest only once?

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Correct Ans:360

Explanation:

The given digits are 4, 4, 6, 6, 6, 7 and 0. Totally we have 7 digits. So numbers greater than a million can be formed by using all the digits.
4 occurs twice, 6 occurs thrice while 0 and 7 once,
Therefore total number of arrangements = 7! / (3! x 2!) = 420
We have to avoid 0 in the starting place, Number of ways in which 0 can be in the first place is 6! / 2! X 3! = 60
Hence the total number of ways = 420 – 60 = 360

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18. In how many ways a four digit even number can be formed by using the digits 4,5,9,8 exactly once.

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Correct Ans:12

Explanation:

**Given**

Four digit even number can be formed by using the digits 4,5,9,8

Since the number has to be a even digit number,the units digit has to be 4 or 8. First three places can be filled by remaining three digits.

Hence it is totally 6 x 2 = 12 ways

**Answer is**12

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19. In how many ways a four digit even number can be formed by using the digits 2,3,5,8 exactly once?

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Correct Ans:12

Explanation:

**Solution is**

Given

Given

Four digit even number can be formed by using the digits 2,3,5,8

Since the number has to be an even number,the unit digit has to be 2 or 8.

First three places can be filled by remaining three digits.

= 3! + 3!

= (3*2*1)+(3*2*1)

= 6 + 6 =12

Hence totally

**12 ways**

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20. In a plane there are totally 8 points (no three points are collinear), how many lines can be drawn ?

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Correct Ans:28

Explanation:

To draw a line we need 2 points.
From the given 8 points , we need to choose 2 points
Number of ways in which this can be done is 18C2= (8 x 7 ) / 2 = 28

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