Factorial (Practice Problems)

Last Updated : 19 Sep, 2026

Factorial is the product of all positive integers less than or equal to a given number. It is represented by the symbol (!).

Solved Questions

Question 1. Evaluate the following.

  • Factorial of 1
  • Factorial of 3
  • Factorial of 4
  • Factorial of 6
  • Factorial of 7
  • Factorial of 8
  • Factorial of 9

Solution:

Factorial of 1 = 1! = 1

Factorial of 3 = 3! = 3 × 2 × 1 = 6

Factorial of 4 = 4! = 4 × 3 × 2 × 1 = 24

Factorial of 6 = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Factorial of 7 = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Factorial of 8 = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320

Factorial of 9 = 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 =362880

Question 2. What is the value of factorial: 14! / (11! × 4!)

Solution:

14! / (11! × 4!) = (14 × 13 × 12 × 11!) / (11! × 4!)

⇒ 14! / (11! × 4!) = (14 × 13 × 12) / 4!

⇒ 14! / (11! × 4!) = (14 × 13 × 12) / (4 × 3 × 2 × 1!)

⇒ 14! / (11! × 4!) = (14 × 13 × 12) / (12 × 2 )

⇒ 14! / (11! × 4!) = (7 × 13)

⇒ 14! / (11! × 4!) = 91

Question 3. Evaluate the expression 6! - 3!

Solution:

6! - 3! = (6 × 5 × 4 × 3!) - 3!

⇒ 6! - 3! = (6 × 5 × 4 × 3!) - 3!

⇒ 6! - 3! = (120 × 3!) - 3!

⇒ 6! - 3! = 3![120 - 1]

⇒ 6! - 3! = 6 × 119

⇒ 6! - 3! = 714

Question 4. If (1 / 6!) = (x / 8!) - (1 / 7!), then find the value of x.

Solution:

(1 / 6!) = (x / 8!) - (1 / 7!)

⇒ (1 / 6!) = (x / 8 × 7!) - (1 / 7!)

⇒ (1 / 6!) = (1 / 7!)[(x / 8) - 1]

⇒ (1 / 6!) = {1 / (7 ×6!)}[(x / 8) - 1]

⇒ (1 / 6!) = (1 / 6!)(1 / 7 )[(x / 8) - 1]

⇒ 1 = (1 / 7 )[(x / 8) - 1]

⇒ 7 = (x / 8) - 1

⇒ (x / 8) = 7 + 1

⇒ (x / 8) = 8

⇒ x = 64

Question 5. How many 4-digit numbers can be formed using the digits 4,6,7,9 in each of which no digit is repeated?

Solution:

Given:

Digits: 4, 6, 7, and 9

Number of digits = 4

We have to arrange these digits to form a 4-digit number.

The number of ways for arranging these digits to form a 4-digit number is 4!

and 4! = 4 × 3 × 2 × 1 = 24

Thus, there are 24 ways in which a 4 digit number can be formed without repeating the digits.

Question 6. Evaluate the expression 3! (2! × 0!)

Solution:

3! (2! × 0!) = (3 × 2 × 1) (2 × 1 × 1) [By using factorial formula and 0! = 1]

⇒ 3! (2! × 0!) = 6 × 2

⇒ 3! (2! × 0!) = 12

Practice Questions

Problem 1: Evaluate.

  • (8! × 7!) / 6!
  • 7! / 4!
  • 10! − 9!

Problem 2: Simplify.

  • (7 + 3)! / 2!
  • 6! / (4! × 2!)
  • (9!) / [(7!) × (2!)]
  • (6!) / [(5!) × (3!)]
  • (12!) / [(11!) × (10!)]

Problem 3: Find the Value of n if

  • n! = 120
  • (n − 1)! = 24
  • (n + 2)! = 720
  • (n − 2)! = 120

Problem 4: If n! / (n−3)! = 120, find the value of n.

Problem 5: Prove that n! is divisible by (n−k)! for all integers n and k such that 0 ≤ k ≤ n.

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