**Permutation And Combination**

Preparing for competitive exam and don’t know what the** Permutation And Combination** is? Then don’t be panic, it is the interesting part of Maths… In this article, we’re going to talk about the basic concepts of Permutation & Combination and formulas / Shortcuts required for solving problems and get the answers of related questions. One must go through this article and will become well-known with concepts of permutation and combination and prepare well.

In all competitive exams, the topic of Permutation and Combination is also covered, so you cannot neglect it. One must learn the Permutation and Combination Concepts, Permutation and Combination Basics and for this concern we’re creating this article for you. Those, who’re going to appear in any competitive exam, they must go through this article of recruitmentresult.com, take a look…

__What Is Permutations And Combinations__?

Ever got confused between which one is what? So don’t be confused and check Permutation and Combination Meaning first and the basic Permutation And Combination Difference is that Permutation denotes arrangement of objects in particular order while Combination represents selecting things at random (order doesn’t matters).

__Permutation And Combination Formula__?

In this section, we’re providing you Permutation And Combination Aptitude Tricks, Permutation And Combination Shortcuts and Formulas to solve the question easily, take a look…

__What is Factorial Notation__?

Let n be a positive integer, then, factorial n, denoted n! Is defined as: n! = n (n – 1) (n – 2) … 3.2.1.

__For Examples__:

- We define 0! = 1.
- 4! = (4 x 3 x 2 x 1) = 24.
- 5! = (5 x 4 x 3 x 2 x 1) = 120.

### Permutation And Combination

Basically the Permutation is of two types namely **Repetition is Allowed** & **No Repetition**. Now one must check from here the formulas for solving both the type question:

__Permutations with Repetition__

These are the easiest to calculate or we can say that the different arrangements of a given number of things by taking some or all at a time.

__For Examples__:

- All permutations (or arrangements) made with the letters x, y, z by taking two at a time are (xy, yx, xz, zx, yz, zy).
- All permutations made with the letters x, y, z taking all at a time are: (xyz, xzy, yxz, yzx, zxy, zyx)

Check Out: __How to Prepare for Maths__

__Permutations without Repetition__

The Number of all permutations of n things, taken r at a time, is given by the following formula as shown below:

__For Example__:

^{6}P_{2}= (6 x 5) = 30^{7}P_{3}= (7 x 6 x 5) = 210

__Combination with Repetition__

In this term, each of the different groups or selections can be formed by taking some or all of a number of objects.

__For Examples__:

Suppose we want to select two out of three boys X, Y, Z then, possible selections are XY, YZ and ZX.

__Combination without Repetition__

The number of all combinations of n things, taken r at a time is as calculated.

__Essential Note__:

^{n}C_{n}= 1 and^{n}C_{0}= 1.^{n}C_{r}=^{nC}(n – r)

__Permutation And Combination Examples__:

^{11}C_{4 }= (11 x 10 x 9 x 8) / (4 x 3 x 2 x 1) = 330

__Keep An Eye On__:

- Fundamental Principles of Counting: If there are x ways to do one task, and y ways to do one more, then there are x × y ways of doing both and the Fundamental Counting rule is the guiding rule for finding the number of ways to complete two tasks.
- Principle of Addition: If an event can happen in ‘x’ ways and another event can occur in ‘y’ ways sovereign of the first event, then either of the two events can occur in (x + y) ways.
- Principle of Multiplication: If an operation can be executed in ‘x’ ways and after it has been achieved in any one of these ways, a second operation can be executed in ‘y’ ways, then the two operations in sequence can be executed in (x × y) ways.

__Permutation And Combination Basics __

Check from here the Permutation And Combination Formula Sheet to solve the Permutation And Combination Sums…

- n!=n×(n−1)!
- 0!=1!=1

__Permutation And Combination Tricks__:

From this section, you check the Permutation And Combination Aptitude Shortcuts and Permutation And Combination All Formulas to solve Permutation And Combination Problems, take a look…

- No of permutations of n different things, taken r at a time, when a particular thing is to be always included in each arrangement is
^{r}n−1*P_{r−1} - No of permutations of n different things, taken r at a time, when a particular thing is never taken in each arrangement is
^{n−1}P_{r} - No of permutations of n different things, taken all at a time, when m specified things always come together is m!*(n−m+1)!
- No of permutations of n different things, taken all at a time, when m specified never come together is n!−[m!*(n−m+1)!]
- The no of permutations of n dissimilar things taken r at a time when k (< r) particular things always occur is [
^{n−k}P_{r−k}]×[^{r}P_{k}] - The no of permutations of n dissimilar things taken r at a time when k particular things never occur is
^{n−k}P_{r} - The no of permutations of n dissimilar things taken r at a time when repetition of things is allowed any number of times is n
^{r} - The no of permutations of n different things, taken not more than r at a time, when each thing may occur any number of times is n+n2+n3+….+n
^{r}=n(n^{r}−1) / n−1

Start Now! __Mathematics Quiz__

__Permutation And Combination Problems And Solutions__

Check the solved Permutation And Combination Questions from here for different competitive exams.

Ques 1. In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together?

- 120
- 720
- 4320
- 2160
- None of these

Answer: B

Permutation And Combination 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.

Ques 2: In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

- 63
- 90
- 126
- 45
- 135

Answer: A

Ques 3: 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?

- 40
- 400
- 5040
- 2520

Answer: C

Explanation:

‘LOGARITHMS’ contains 10 different letters.

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.

Ques 4: From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

- 564
- 645
- 735
- 756
- None of these

Answer: D

Ques 5: In how many different ways can the letters of the word ‘LEADING’ be arranged in such a way that the vowels always come together?

- 360
- 480
- 720
- 5040
- None of these

Answer: C

Explanation:

The word ‘LEADING’ has 7 different letters.

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

Then, we have to arrange the letters LNDG (EAI).

Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.

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

Required number of ways = (120 x 6) = 720.

Ques 6: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

- 210
- 1050
- 25200
- 21400
- None of these

Answer: C

Ques 7: In how many different ways can the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together?

- 810
- 1440
- 2880
- 50400
- 5760

Answer: D

Explanation:

In the word ‘CORPORATION’, we treat the vowels OOAIO as one letter.

Thus, we have CRPRTN (OOAIO).

This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.

Number of ways arranging these letters = 7!/2! = 2520.

Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in

5!/3! = 20 ways.

Required number of ways = (2520 x 20) = 50400.

Ques 8: In how many ways can the letters of the word ‘LEADER’ be arranged?

- 72
- 144
- 360
- 720
- None of these

Answer: C

Ques 9: How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

- 5
- 10
- 15
- 20

Answer: D

Explanation:

Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

Required number of numbers = (1 x 5 x 4) = 20.

Ques 10: In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

- 159
- 194
- 205
- 209
- None of these

Answer: D

Ques 11: In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women?

- 266
- 5040
- 11760
- 86400
- None of these

Answer: C

Ques 12: 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?

- 32
- 48
- 36
- 60
- 120

Answer: C

**Permutation And Combination For Cat**

Ques 13: A five-digit number is formed using digits 1, 3, 5, 7 and 9 without repeating any one of them. What is the sum of all such possible numbers?

- 6666600
- 6666660
- 6666666
- None of these

Answer: A

Ques 14: How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the unit’s place must be greater than that in the ten’s place?

- 54
- 60
- 17
- 2 × 4!

Answer: B

Ques 15: How many numbers can be made with digits 0, 7, 8 which are greater than 0 and less than a million?

- 496
- 486
- 1084
- 728

Answer: D

Above provided Permutation and Combination Definition & Permutation and Combination Formula List will helpful for you to solve any Permutation and Combination Difficult Questions. One must go through this article to get more information about Permutation And Combination Books, How to Solve Permutation and Combination? When To Use Permutation And Combination In Probability? Permutation And Combination Examples With Answers. Students may also sharpen their skills by giving the Permutation And Combination Online Test.

As per recent survey we get the information that the Permutation and Combination Class 11 Exam Fear is found in many students. So don’t be hopeless and get complete details about Permutation And Combination Tutorial, Permutation And Combination Notes, Permutation And Combination Class 11 Ncert Solutions, Permutation And Combination Questions And Answers Pdf and Permutation And Combination Practice Problems from this article. Practice through above listed Permutation And Combination Gre / IIT JEE Problems and do well in examination. All The Best.

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