- A person has 12 friends out of which 7 are relatives. In how many ways can he invite 6 friends such that at least 4 of them are relatives?
- In a party 19 persons shake hands with every other person. Find the total no of handshakes in the party?
- In how many ways 6 novels and 7 notebooks can be arranged in a row so that they are alternate?
- Geography books are always together.
- History books are never together.
- Geography books are never together.
- History books are always together and also Geography books are always together?
- How many four digit numbers greater than 5000 can be formed with the digits 4 ,5, 7, 8, and 9?
- 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?
- 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?
- 8 students appear in an examination. In how many ways can the result be announced?
- How many 8 digit numbers can be formed using the digits 1, 2, 0, 2, 4, 1, 2, 4 ?
- How many words can be formed with the letter of the word ‘NUMBER’ when alll 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?
- In how many different ways can the letters of the word ‘PADDLED’ be arranged?
- In how many different ways can the letter of the word ‘STRONG’ be arranged?
- 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...
- In how many different ways can the letter of the word ‘PRAISE’ be arranged?
- In how many different ways can 6 boys and 3 girls to be seated in a row such that all the boys seated together and all the girls seated together?
- In how many different ways can the letter of the word ‘JUSTICE’ be arranged?
- In how many different ways can the letters of the word “INCORPORATION” be arranged so that the vowels comes together?
- From 6 officers and 8 Jawans, in how many ways can 5 be chosen to include exactly 3 officers?
- How many words can be formed from the word INCREASE in which both E’s do not come together?
- In how many different ways can the letters of the word “MONDAY” be arranged in such a way that the vowels always come together?
- In how many different ways can the letters of the word ‘VARIOUS’ be arranged in such a way that the vowels occupy only the odd positions? ...
- In how many ways can a group of 2 boys and 2 girls be made out of a total of 6 boys and 4 girls?
- In how many ways Barcelona’s football team captain and vice-captain be elected from a team of 15 men?
- There are 10 members in a delegation which is to be sent at SAARC summit. The total number of members is 18. In how many ways can the selection be made so ...
- 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?
- Find the no. of words formed by using all the letter of the word DISCOUNT, so that the vowels are never together?
- A Consignment of 12 mobile phones contains 4 defectives. The mobile phones are selected at random one by one and examined. The ones examined are not put ba...
- Akshay has 7 relatives, 3 males and 4 females. His wife twinkle also have 7 relatives, 3 females and 4 males. In how many ways can they invite 3 male and 3...
- Maria and Ariana were only two girls participating in a Billiards tournament. Every Participant played two games with every other participant. The number o...
- In how many ways a Cricketer can make a Double century with fours and Sixes only?
- How many different nine digit numbers can be formed from the number 66 55 99 222 by rearranging its digits so that the odd digits occupy even positions onl...

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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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