Factorial : Examples

---

Example :  

Simple                                     Factorial Rules : Click Here

Q1. How many permutations can be formed from the letters of the word “SMILE”?

Solⁿ

5! = 5 × 4 × 3 × 2 × 1 = 120, (All letters distinct)

Ans: 120                   

Q2. How many permutations can be made using the letters of “GOA”?

Solⁿ

3! = 3 × 2 × 1 = 6, (All letters distinct)

Ans: 6

Medium                                             Factorial Rules : Click Here

Q3. How many different words can be formed from “INDIA”?

Solⁿ

Permutations = 5!/2! = 120/2 = 60, (I repeats 2 times)

Ans: 60

Q4. How many permutations are possible from the letters of “LEVEL”?

Solⁿ

5! / 2! × 2! = 1204 = 30, (L repeats 2 times, E repeats 2 times)

Ans: 30

Tough                                                     Factorial Rules : Click Here

Q5. How many permutations can be formed from the word “BALLOON”?

Solⁿ

L repeats 2 times, O repeats 2 times

7! / 2! × 2! = 5040/4 = 1260

Ans: 1260

Q6. How many distinct permutations can be made from “COMMITTEE”?

·         M = 2 repeats

·         T = 2 repeats

·         E = 2 repeats

Solⁿ

9! / 2! × 2! × 2! = 362880/8 = 45360

Ans: 45360

Q1. How many distinct arrangements (words) can be formed using the letters of the word “INDIA”?

a. 120   b. 60   c. 30   d. 24

Ans: b. 60

Explanation: Letters in I,N,D,I,A= 5, Letter I repeats 2 times

Total arrangements = 5!/2! = 120 /2 = 60

Q2. How many different permutations can be formed using all letters of the word “DOCTOR”?

a. 720  b. 360  c. 180  d. 120

Ans: b. 360                                                 Factorial Rules : Click Here

Explanation: Letters in D,O,C,T,O,R = 6, Letter O repeats 2 times

Total arrangements = 6!/2! = 720/2 = 360

Q3. How many distinct arrangements (words) can be made from the letters of “SMILE”?

a. 24  b. 60  c. 120  d. 720

Ans: c. 120

Explanation: All 5 letters are different

5! = 120

Q4. Which word has repeated letters ?

a. SMILE  b. DOCTOR  c. INDIA  d. Both b and c

Ans: d. Both b and c

Explanation:

·         DOCTOR → O repeats

·         INDIA → I repeats

·         SMILE → no repetition                                                             Factorial Rules : Click Here

Q5. If all letters are distinct in a word of 5 letters, how many permutations can be formed ?

a. 120  b. 60  c. 720  d. 24

Ans: a. 120

Explanation: 5! = 120                

Q. In how many different ways can the letters of the word "FACEBOOK" be arranged ?

A) 12160  B) 26342  C) 20160  D) 10080

Solution

The word FACEBOOK has 8 letters: F, A, C, E, B, O, O, K

• Total letters = 8


• The letter O is repeated 2 times.

Formula : Number of arrangements = n! / r! ​

where: n = 8 , r = 2

= 8! / 2! = 40320 / 2 = 20160

​=20160

Ans: C) 20160

Tricks for Competitive Exams

1st: Count the total letters (8).

2nd: Check for repeated letters (O appears twice).

3rd: Use the formula:

8! / 2! = 40320 / 2 = 20160

Shortcut Formula

• No repeated letters → n!


• One letter repeated twice → n! / 2!​


• One letter repeated three times → n!/3! ​


• Multiple repeated letters → n! / p! × q! × r!​

Ans: C) 20160

Q) In how many different ways can the letters of the word "BIHAR" be arranged ? প্ৰশ্ন: "BIHAR" শব্দটোৰ আখৰসমূহক কিমান ধৰণে সজাব পাৰি ?

A) 60   B) 120  C) 180  D) 240

Solution | সমাধান

The word BIHAR has 5 letters. ("BIHAR" শব্দটোত ৫টা আখৰ আছে।)

B, I, H, A, R

• All the letters are different. (সকলো আখৰ বেলেগ-বেলেগ।)


• No letter is repeated. (এটা আখৰো পুনৰাবৃত্তি হোৱা নাই।)

Formula | সূত্ৰ : If all letters are different: Number of arrangements = n!​ (যদি সকলো আখৰ বেলেগ হয়: সজোৱাৰ সংখ্যা = n!)​

Here, 5! = 5 × 4 × 3 × 2 × 1 = 120

Ans | সঠিক উত্তৰ: B) 120

Trick | ট্ৰিক

• All letters different (সকলো আখৰ বেলেগ হলে) → n!


• One letter repeated twice (এটা আখৰ ২ বাৰ থাকিলে) → n! / 2!​


• One letter repeated three times (এটা আখৰ ৩ বাৰ থাকিলে) → n! / 3! ​​

Ex | উদাহৰণ

• BIHAR → 5! = 120


• BOOK → 4! / 2! = 12


• FACEBOOK → 8! / 2! = 20160