# How To Solve Quickly Permutation Combination

## How to Solve Permutation and Combination Questions Quickly.

**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^{n}P_{r}= 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^{n}C_{r}= n!/(r!) (n-r)!

Here we can easily understand how to solve permutation and combination easy.

**Read Also –****Formulas to solve permutation questions**

## 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

OR

We can solve directly by formula n^{r} = 6^{3} = 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:

5!/2! * 2!

= 5 * 4 * 3 * 2 * 1 / 2 * 1 * 2 * 1

= 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

OR

We can solve directly by formula n^{r} = 5^{3} = 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:

^{10}P_{5} = 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}C_{15} =^{19}C_{15}

We know that, ^{n}C_{r} = n!/(r!) (n-r)!^{19}C_{15} = 19!/15! (19 – 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}C_{10} = ^{14}C_{10}

We know that, ^{n}C_{r} = n!/(r!) (n-r)!^{14}C_{10} = 14!/10! * (14 -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

Correct option: C

#### Solution:

^{r + n – 1}C_{r} = ^{16 + 4 – 1}C_{16} = ^{19}C_{16}

We know that, ^{n}C_{r} = n1/(r!) (n-r)!^{19}C_{16} = 19!/16! * (19 -16)! = 969

#### Correct option: B

## 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. 2520

B. 1200

C. 210

D. 720

#### Solution:

Number of ways of selecting (5 consonants out of 10) and (2 vowels out of 4) = 10C5 * 5C2 = 252 * 10 = 2520

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