# 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: 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 (^{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. 2520**

**B. 1200**

**C. 210**

**D. 720**

**Solution: **Number of ways of selecting (5 consonants out of 10) and (2 vowels out of 4) = ^{10}C_{5} * ^{5}C_{2} = 252 * 10 = 2520

**Correct option: A**

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

Very interesting questions & helps to understand d concept

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

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