How To Solve Quickly Permutation Combination

July 27, 2023

How to Solve Permutation and Combination Questions Quickly

How to solve Permutation and Combination Questions Quickly has been discussed on this page along with basic concepts.

What is Permutation? Permutation is basically called as a arrangement where order does matters.Here we need to arrange the digits , numbers , alphabets, colors and letters taking some or all at a time.It is represented as,

\mathbf{^nP_r = \frac{n!}{(n-r)! }}

What is Combination? Combination is basically called as a selection where order does not matters.Here we need to arrange the digits , numbers , alphabets, colors and letters taking some or all at a time. It is represented as

\mathbf{^nC_r = \frac{n!}{(n-r)! r! }}

Basic concept of Permutation and Combination

Permutation is an arrangement of objects in a definite order.
Number of all permutations of n things, taken r at a time, is given by
ⁿP_r = $\mathbf{\frac{n!}{(n-r)!}}$
Combination is selection of objects where order does not matter.
Number of all combinations of n things, taken r at a time, is given by
ⁿC_r = $\frac{n!}{(n-r)! r! }$
Here we can easily understand how to solve permutation and combination easy.

Type 1: How to Solve Quickly Permutation and Combination Different ways to arrange (with repetition)

Question 1.How many 3 letter words with or without meaning can be formed out of the letters of the word MONDAY when repetition of words is allowed?

Options:

A. 125

B. 216

C. 120

D. 320

Solution: 6 * 6 * 6 = 216

We can solve directly by formula n^r = 6³ = 216

Correct option: B

Question 2. In how many ways the letters in the word TOOTH can be arranged?

Options:

A. 120

B. 40

C. 20

D. 30

Solution: $\frac{5!}{2! × 2! }$

= $\frac{5 × 4 × 3 × 2 × 1 }{2 × 1× 2 × 1 }$

= $\frac{120}{ 4}$

= 30

Correct option: D

Question 3.How many three digit numbers can be formed using digits 2, 3, 4, 7, 9 so that the digits can be repeated.

Options:

A. 125

B. 360

C. 24

D. 6

Solution: Each place can be filled by any one of 5 digits

Total numbers = 5 * 5 *5 =125

We can solve directly by formula n^r = 5³ = 125

Correct option: A

Type 2: Different ways to arrange (without repetition)

Question 1. How many five letter words with or without meaning, can be formed from the word ‘COMPLEXIFY’, if repetition of letters is not allowed?

Options:

A. 43200

B. 30240

C. 12032

D. 36000

Solution: ¹⁰P₅ = $\frac{10!}{(10 – 5)!}$ = 10 * 9 * 8 * 7 * 6 = 30240

Correct option: B

Question 2.In how many different ways can the letters of the word ‘LOGARITHMS’ be arranged so that the vowels always come together?

Options:

A. 6720

B. 241920

C. 40320

D. 360344

Solution: In such questions we treat vowels as one letter.

So the word becomes LGRTHMS (OAI)

It means there are total 8 letters. Therefore, number of ways of arranging these letters = 8! = 40320

Now, there are three vowels (OAI), number of ways of these letters can be arranged = 3! = 6

Required number of words = 40320 * 6 = 241920

Correct option: B

Question 3.How many three digit numbers can be formed from the digits 3, 4, 5, 7, 8, and 9. Also, the number formed should be divisible by 5 and no repetition is allowed?

Options:

A. 20

B. 24

C. 25

D. 10

Solution: The number which is divisible by 5 has 5 or 0 at one’s place. In this case we must have 5 at the unit place as 0 is not in the list.

There are total 6 digit out of which last digit is fixed by 5. Therefore, we are left with 5 digits (3, 4, 7, 8, 9) at the tens place.

Similarly, the hundred place can be filled by 4 digits.

So, required number = 4 * 5 * 1 = 20

Correct option: A

Type 3: How To Solve Permutation and Combination Question- (with repetition)

Question 1.An ice cream seller sells 5 different ice-creams. John wants to buy 15 ice creams for his friends. In how many ways can he buy the ice-cream?

Options:

A. 1450

B. 3768

C. 3876

D. 1540

Solution: ^{r + n -1}C_r = ^{15 + 5 – 1}C₁₅ =¹⁹C₁₅

We know that, ⁿC_r = $\frac{n!}{(n-r)! r! }$

¹⁹C₁₅ = $\frac{19!}{(19-15)! 15! }$ = 3876

Correct option: C

Question 2.There are 5 types of soda flavor available in a shop. In how many ways can 10 soda flavors be selected?

Options:

A. 1454

B. 1001

C. 1211

D. 1540

Solution:^{r + n – 1}C_r = ^{10 + 5 – 1}C₁₀ = ¹⁴C₁₀

We know that, $\frac{n!}{(n-r)! r! }$

¹⁴C₁₀ = $\frac{14!}{(14-10)! 10! }$ = 1001

Correct option: B

Question 3. In how many ways can 16 identical toys be divide in 4 children?

Options:

A. 966

B. 696

C. 969

D. 996

Solution: ^{r + n – 1}C_r = ^{16 + 4 – 1}C₁₆ = ¹⁹C₁₆

We know that, ⁿC_r = $\frac{n!}{(n-r)! r! }$

¹⁹C₁₆ = $\frac{19!}{(19-16)! 16! }$ = 969

Correct option: C

Type 4: Permutation and Combination Solve Question Quickly. (without repetition)

Question 1. A wooden box contains 2 grey balls, 3 pink balls and 4 green balls. Fins out in how many ways 3 balls can be drawn from the wooden box. Make sure that at least one pink ball is included in the draw?

Options:

A. 64

B. 46

C. 56

D. 65

Solution: According to the question, we have, (one pink and two non-pink balls) or (two pink and one non-pink balls) or (3 pink)

Therefore, required number of ways are (³C₁ * ⁶C₂) + (³C₂ * ⁶C₁) + (³C₃) = 45 +18 + 1 = 64

Correct option: A

Question 2.There are 5 boys and 10 girls in a classroom. In how many ways teacher can select 2 boys and 3 girls to make a dance group?

Options:

A. 720

B. 1200

C. 240

D. 840

Solution: Required numbers of ways = ⁵C₂ * ¹⁰C₃ = 10 * 120 = 1200

Correct option: B

Question 3.There are 10 consonants and 5 vowels. Out of which how many words of 5 consonants and 2 vowels can be made?

Options:

A. 1270080

B. 120052

C. 210789

D. 720432

Solution: Number of ways of selecting (5 consonants out of 10) and (2 vowels out of 4) = ¹⁰C₅ * ⁵C₂ = 252
Number of ways of arranging 7 letters among themselves = 7!

=7 x 6 x 5 x 4 x3 x2 x 1

=5040

Required number of ways = (252 x 5040) = 12,70,080

Correct option: A

Prime Course Trailer

Related Banners

Get PrepInsta Prime & get Access to all 200+ courses offered by PrepInsta in One Subscription

Get over 200+ course One Subscription

Courses like AI/ML, Cloud Computing, Ethical Hacking, C, C++, Java, Python, DSA (All Languages), Competitive Coding (All Languages), TCS, Infosys, Wipro, Amazon, DBMS, SQL and others

Checkout list of all the video courses in PrepInsta Prime Subscription

Probability – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts

Others

Combination – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts
Circular Permutation – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts

Probability –
Questions |
Formulas |
How to Solve Quickly |
Tricks & Shortcuts

Others

Combination –
Questions |
Formulas |
How to Solve Quickly |
Tricks & Shortcuts
Circular Permutation –
Questions |
Formulas |
How to Solve Quickly |
Tricks & Shortcuts

10 comments on “How To Solve Quickly Permutation Combination”

Rohit
very clear… so much helpful

Log in to Reply
Sameer
very helpful

Log in to Reply
Esha
This is a very well framed site to help everything better , really like it

Log in to Reply
Kishore
type 2 questions were new to me.. thanks alot

Log in to Reply
Prabh
Very interesting questions & helps to understand d concept

Log in to Reply