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Q1(i):
Evaluate (i) $8!$
Solution :
Step 1: Theoretical Definition of Factorial Notation
The factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers from $1$ up to $n$. Mathematically, it is defined as:
$n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1$
[Per the definition of the factorial function in combinatorial mathematics].
Step 2: Expansion of 8!
Applying the definition to $n = 8$, we expand the factorial as the product of integers in descending order until reaching unity:
$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Step 3: Sequential Calculation
To ensure numerical accuracy, we perform the multiplication in cumulative stages:
- $8 \times 7 = 56$
- $56 \times 6 = 336$
- $336 \times 5 = 1,680$
- $1,680 \times 4 = 6,720$
- $6,720 \times 3 = 20,160$
- $20,160 \times 2 = 40,320$
- $40,320 \times 1 = 40,320$
Visual Representation of Factorial Growth
Summary of Calculation
By evaluating the product of the first eight natural integers, we arrive at the total number of permutations of eight distinct objects, which is $40,320$.
Final Solution: $8! = 40,320$
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.2
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
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