Factorial Notations.pptx

Factorial Notations

1. Factorial Notations

2. The factorial of a number is the function that multiplies the number by every  natural number  below it. Symbolically, factorial can be represented as "!". So,  n  factorial is the product of the first  n  natural numbers and is represented as n! For example, 4 factorial, that is, 4! Can be written as: 4! = 4×3×2×1= 24 Introduction

3. The factorial notation comes in handy when you are arranging objects. Consider the following scenario that we shall use to use to define and introduce this notation. For example, you have ten balls. Each ball has a number marked on it. You also have ten slots that you have to fill with the balls. How many different ways can you fill these slots in?

4. n! = n*(n -1)! Formula

5. Example 1 : Q. What is the factorial of 5 ? Ans : We know that the factorial formula is
n! = n × (n – 1) × (n – 2) × (n – 3) × ….× 3 × 2 × 1
So the factorial of 5 is
5! = 5 × (5 -1) × (5 – 2) × (5 – 3) × 1
5! = 5 × 4 × 3 × 2 ×1
5! = 120
Therefore, the factorial of 6 is 120. Example 2 : Q. how many ways can you arrange the letters in the word “Bishal” without repeating them? Ans: For this problem, count the number of letters in the word “Bishal” to find there are six letters. Then, find the factorial of the number six . n! = n(n-1)! 6! = 6(6 − 1)(6 − 2)(6 − 3)(6 − 4)(6 − 5) 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 Now you know that the maximum number of ways, you can arrange the letters in the word“Bishal” with no repeats is 720.

6. Use of Factorial One area where factorials are commonly used is in permutations & combinations. Now, permutation is an ordered arrangement of outcomes and it can be calculated with the formula: Combination is a grouping of outcomes in which the order does not matter. It can be calculated with the formula:

7. Example 1 : Q. In a group of 10 people, $200, $100, and $50 prizes are to be given. In how many ways can the prizes be distributed? Ans : This is permutation because here the order matters. It can be calculated as 10P3 ways. Example 2 : Q. Three $50 prizes are to be distributed in a group of 10 people. In how many ways can the prizes be distributed? Ans : This is a combination because here the order does not matter. It can be calculated as, 10C3 ways.

8. Factorial of 0 Mathematicians agree that the factorial for the number zero is one, or 0! =1. It might seem odd that 0! =1, but it’s easy to understand if you follow the pattern of factorials backward. Look at this pattern starting with 4!: 4! = 24, 3! = 6, 2! = 2, 1! = 1, 0! = 1 We can notice that each answer is divisible sequentially and as you follow the pattern, it predicts the next answer and shows that 0! =1. The sequential divisible numbers are in bold: 4! = 24, (24 ÷ 4 = 6) 3! = 6, (6 ÷ 3 = 2) 2! = 2, (2 ÷ 2 = 1) 1! = 1, (1 ÷ 1 = 1) 0! = 1

9. Factorial of negative number We cannot find the factorial for a negative number. To find the factorial for a negative integer, you would have to divide by zero. However, dividing by zero is undefined. Therefore, negative integer factorials are undefined.

10. Applications In  mathematical analysis , factorials are used in  power series for the  exponential function  and other functions, and they also have applications in  algebra ,  number theory ,  probability theory , and  computer science .