# How To Solve Quickly Permutation Combination

## How to solve Permutation and Combination Quickly

**Permutation formula used for selection and arrangement of items,\mathbf{^nP_r = \frac{n!}{(n-r)! }} **

**Combination formula used for selection of items,\mathbf{^nC_r = \frac{n!}{(n-r)! r! }} **

### 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}= \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^{n}C_{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

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: ** \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

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} = \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}C_{15} =^{19}C_{15}

We know that, ^{n}C_{r} = \frac{n!}{(n-r)! r! }

^{19}C_{15} = \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}C_{10} = ^{14}C_{10}

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

^{14}C_{10} = \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}C_{16} = ^{19}C_{16}

We know that, ^{n}C_{r} = \frac{n!}{(n-r)! r! }

^{19}C_{16} = \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 (^{3}C_{1} * ^{6}C_{2}) + (^{3}C_{2} * ^{6}C_{1}) + (^{3}C_{3}) = 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 = ^{5}C_{2} * ^{10}C_{3} = 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) = ^{10}C_{5} * ^{5}C_{2} = 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**

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

very helpful

This is a very well framed site to help everything better , really like it

type 2 questions were new to me.. thanks alot

Very interesting questions & helps to understand d concept

Thanking You and keep supporting us by which we will give you the best

these questions are really helps to understands the each and every concepts thank you prep ins teams keep it up

welcome Prakash Kumar

sir provide more question for practice

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