**What is Factorial of a number n, denoted as n!?**

Factorial of any number n is denoted as n! Factorial of a number is the product of all the positive integers from 1 up to n, (including n) (n >= 1)

For example A Useful Tip.

Use the tip to simplify:

Let us now define n! (Factorial of a number n)

Consider

**Now, what does 24 signify?**

Without allowing any letter to repeat, 24 permutations (arrangements) can be formed taking all 4 different things at a time. Caught the point? No? Then Read on

**How many ways can you seat four men a, b, c, d in four chairs?**

Treat the four chairs as the blanks below: —– —– —– —–. Any one of 4 persons can sit in the first chair; any 3 in the 2nd; any 2 in the 3rd; and 1 in the last chair. Filling each chair with a man is one task. And, there are four tasks here. From rule of counting, the four tasks can be completed in 4.3.2.1 ways, i.e., **24**

Each of the 24 ways is an arrangement or more popularly a permutation. So, what does 4! signify? 4! stands for 24 arrangements. Another way to define 4! is. Without allowing things to repeat, the number of ways of arranging 4 different things is 4!. Generalize 4!. How? As follows: Definition of n! Without allowing things to repeat, taking all in each permutation, n different things can be arranged in n! ways. Note: 0! = 1

**Solved Examples:**

**1. Use a factorial formula to simplify:**

**Solution:** in the given example write

*10!-8!/8!-7!=8!(10.9-1)/7!(8-1)=8.7!(89)/7!.7=8.7!(89)/7!.7=89/7*

**2. Express 6.7.8.9.10 in factorial notation**

**Solution:**

*10.9.8.7.6=10.8.9.7.6.5.4.3.2.1/5.4.3.2.1=10!/5!*