- How many numbers of five digits may be formed with the digits 5, 0, 9, 0, and 6?
- In how many ways can 2 girls and 8 boys be seated in a row so that girls are always together?
- Among a set of 4 black balls and 3 white balls, how many selections of 3 balls can be made such that at least 2 of them are black balls?
- An examinee is required to answer the eight questions out of 16 questions which are divided into two groups each containing eight questions each and he is ...
- In an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 8 candidates and 5 members are to be chos...
- How many different words can be formed with the letters of the word PRACTICE such that each of the word begin with E and ends with R?
- A coin is tossed successively three times. Find the probability of getting exactly one tail or two tails?
- In how many ways can the letters of the word ENGINEERING be arranged?
- Salman has a list of 12 friends. He wishes to invite some of them in such a manner that he can enjoy maximum number of parties, but in each party the numbe...
- From 6 officers and 8 Jawans, in how many ways can 5 be chosen to include exactly 3 officers?
- In how many different ways can the letters of the word 'COMPUTER' be arranged in such a way that the vowels always come together?
- How many 10 digit numbers can be formed using the digits 1, 2, 0, 2, 4, 1, 2, 4, 2, 4 ?
- In how many different ways can the letters of the word ‘CURIOES’ be arranged in such a way that the vowels occupy only the odd positions?
- How many five digit numbers can be formed using the digits 0, 1, 2, 3, 4 and 5 which are divisible by 5, without repeating the digits?
- In how many different ways can 4 blue beads and 3 red beads be arranged in a row such that all the blue beads are together and all the red beads are togeth...
- How many no. of 4 digits can be formed with the digits 4, 8, 7 & 3 no digit being repeated. ...
- Find the number of words formed by using all the letter of the word MEMBER, so that the vowels are always together. ...
- There are 14 candidates for 5 post. In how many different ways can be post be filled?
- Out of 6 men and 3 women a committee of 3 members is to be formed so that it has 2 men and 1 woman. In how many different ways can this be done?
- In how many different ways can 6 History and 5Hindi books be arranged in a shelf so that they are arranged alternately?
- In how many different ways can the letters of the word QUANTITY be arranged so that the vowels always come together?
- In how ways can the selection be made so that a particular member is always included? ...
- In how ways can the selection be made so that a particular member is always excluded? ...
- In an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 8 candidates and 5 members are to be chos...
- How many different words can be formed with the letters of the word PRACTICE such that each of the word begin with E and ends with R?
- In how many ways Barcelona’s football team captain and vice-captain be elected from a team of 15 men?
- An examinee is required to answer the eight questions out of 16 questions which are divided into two groups each containing eight questions each and he is ...
- An examinee is required to answer 7 questions out of 14 questions, which are divided into two groups each containing 7 questions. He is not permitted to an...
- In how ways can the selection be made so that a particular member is always included?
- In how ways can the selection be made so that a particular member is always excluded?
- A committee of 5 members is to be formed out of 6 men and 4 women. In how many ways can a committee consisting of atleast 1 woman be formed?
- In how many ways can the letters of the word VOWEL can be arranged so that all the consonants occupy odd places, vowels come on even places?
- How many 8 digit numbers can be formed using the digits 1, 2, 0, 2, 4, 1, 2, 4 ?
- 8 students appear in an examination. In how many ways can the result be announced?
- Find the no. of words formed by using all the letter of the word DISCOUNT, so that the vowels are never together?
- If the letters of the Word SINHA are arranged in all possible ways and these words are written out as in Dictionary, then the rank of the word SINHA is
- In how many different ways can the letters of the word “DISCHARGE” be arranged in such a way that the vowels always come together?
- Among a set of 6 black balls and 3 white balls, how many selections of 5 balls can be made such that at least 3 of them are black balls?
- There are 20 Bikes in the No-Parking zone. On a Particular day, A policeman planned to fine the four bikes of the No-Parking Zone. In how many ways can he ...
- In a Group of 8 boys and 6 girls, Six children are to be selected. In how many different ways can they be selected such that at least one boy should be the...

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Permutation & Combination

Permutation & Combination is an important topic for various competitive exams like IBPS SO, IBPS PO, SBI PO, SBI Clerk, SEBI Grade A, RBI Grade B. Therefore, it is essential to clear the basics of this chapter if you are appearing for any of the above mentioned exams.

Definition of Permutation & Combinations

In Permutation & Combinations we look at various ways in which characters from a given set can be arranged to form subsets without replacements.

In Permutation, an orderly arrangement of elements of a set is involved. Whereas in Combination, we need to look for a number of ways a given set of characters can be arranged without considering their order.

Difference between Permutations and Combinations

Sometimes, the words permutation and combination are used interchangeably. However, they have different implications.

Let’s look at a simple example to understand-

The password of your computer is 1234, but if you enter 4321, it will not unlock even though the numbers are the same. What your computer will recognise is the order of the numbers. There are many possible combinations for the given set of numbers, but your computer accepts only a specific permutation.

This is the basic difference between the two terms.

Difference | Permutation | Combination |
---|---|---|

Application |
Arranging numbers, digits, alphabets, colours, people |
Selection of teams, food, clothes |

Order |
Order matters |
Order does not matter |

Used for |
Lists |
Groups |

Denoted by |
nPr |
nCr |

Permutation

Factorial

Before getting into the basics of this chapter, let us understand what a ‘factorial’ is.

The product of the numbers starting from 1 up to a number ‘n’ is known as the factorial number of ‘n’.

Meaning, n!= 1x2x3x4x5x6…….x(n-2)x(n-1)xn

1!= 1

2!= 1x2= 2

3!= 1x2x3= 6

4!= 1x2x3x4=24

Note1- 0! and 1! Are equal to 1.

Note 2- We cannot find a factorial of a negative number.

Application of factorial

Factorial is most commonly used in arrangements.

For example- We need to arrange 5 persons in a single line. So, we will start with the first place, which means that we can choose 1 person out of the 5 for the first place. This can be done in 5 ways.

Now, 4 places are vacant and 4 people are left. So now, we can choose 1 person out of the 4 for second place. This can be done in 4 ways.

We will repeat the same process for the other 3 places.

To get the final answer we will multiply all these ways for getting the different ways of arrangement.

Therefore, total ways= 5x4x3x2x1 which is 5!= 120

Q1) In how many ways can the letters of the word PATNA be rearranged?

*Answer: *PATNA has a total of 5 words. Therefore, we will arrange 5 letters in 5 places in 5!= 120

However, in this question the letter A has been repeated twice. So, we have to divide by the number of repetitions of the word when any letter appears more than once. In this case there are 2 repetitions.

This means, we have to divide the total 120 ways by 2!= 2

So, the total number of arrangements that can be made= 120/2= 60

There is a way to solve this directly as well= 5!/2!

Combinations

Combinations are relatively easier to solve as the order does not matter here. We can select things at random and check out the different possibilities. Therefore, it is a one step process.

Formula for Combination is nCr= n!/r!* (n-r)!

Let’s look at an example to understand-

In how many ways can a coach choose three players from among five players?

*Answer: *There are 5 players to be taken 3 at a time.

Using the formula:

C(5,3) = P(5,3)/ 3!

= 5×4×3/3×2×1

=10

Thus, the coach can choose the players in 10 different ways.

Formulas for Permutation & Combination

In order to solve question on permutation & combination easily, you need to remember the formulas-

Permutation or Combination | Repetition | Formula |
---|---|---|

Permutation |
Yes |
P (n,r) = nr |

Permutation |
No |
P (n,r) = n! / (n – r)! |

Combination |
Yes |
C (n,r) = n! / r! ( n-r)! |

Combination |
No |
C (n+ r -1 ,r) = (n + r -1 )! / r! (n – 1) ! |

Solved Questions

Q1) In a class there are 4 boys and 5 girls. In how many ways can a class monitor can be chosen?

*Answer:* Here, we have to choose 1 student out of 9 to be the monitor.

So, as per the formula nCr= 9C1= 9/1=9

Q2) How many different words can be made using letters of PATNA starting with P?

*Answer:* PATNA has 5 words. As per the question, P is fixed in the first place. So, we need to arrange the remaining 4 letters at 4 places= 4!= 24 ways.

However, the letter A is repeated twice, so we need to divide the total 24 ways by 2!= 2.

Therefore, different words starting from P= 24/2= 12

Direct approach= 4!/2!= 12

Q3) In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

*Answer:* The word 'OPTICAL' has 7 letters. It has the vowels 'O','I','A' in it and these 3 vowels should always come together. Hence these three vowels can be grouped and considered as a single letter. That is, PTCL(OIA).

Hence we can assume total letters as 5 and all these letters are different.

Number of ways to arrange these letters= 5!= 5x4x3x2x1= 120

All the 3 words (OIA) are different

Number of ways to arrange these vowels among themselves= 3!= 3x2x1= 6

Hence, required number of ways= 120x6= 720

Q4) An urn contains 5 red balls and 3 blue balls. In how many different ways can 2 red and 1 blues balls be drawn?

*Answer:* Ways of selecting 2 red balls= 5C2= 10

Similarly, ways of selecting 1 blue ball= 3C1= 3

So total ways to select 2 red and 1 blue ball= 10*3= 30

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