i need explanation how they solved this one the answer is provided:
finding the number o fdistinguishable permutations.
(15!)/(4!3!612!) = 6, 306,300
I can assure you it is wrong. The denominator with the factor 612! is much greater than the numerator, therefore, the value as written should be less than one. Something is greatly wrong.
yes instead of being 612! needs to be 6!2!
6(x+1)=12(x-3)
To understand how the number of distinguishable permutations was found using the given formula, let's break it down step by step:
Step 1: Understand Permutations
In mathematics, a permutation is an arrangement of objects in a specific order. The number of distinguishable permutations is the count of all possible distinct arrangements of objects without repetition.
Step 2: Identify the Formula
The formula provided for finding the number of distinguishable permutations is:
(15!)/(4!3!612!)
Step 3: Evaluate the Formula
Now, let's evaluate this formula by understanding what each number represents:
"!" denotes the factorial of a number, which means multiplying that number by all the positive whole numbers less than it down to 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
15! means the factorial of 15, which is the product of all positive whole numbers from 15 down to 1.
(15!) = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
4! means the factorial of 4, which is the product of all positive whole numbers from 4 down to 1.
(4!) = 4 x 3 x 2 x 1
3! means the factorial of 3, which is the product of all positive whole numbers from 3 down to 1.
(3!) = 3 x 2 x 1
612! means the factorial of 612, which is the product of all positive whole numbers from 612 down to 1.
(612!) = 612 x 611 x ... x 3 x 2 x 1
Now, substitute the values into the formula:
(15!)/(4!3!612!) = (15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (4 x 3 x 2 x 1) x (3 x 2 x 1) x (612 x 611 x ... x 3 x 2 x 1)
By calculating the numerator and denominator separately, you get:
15! = 1,307,674,368,000
4! = 24
3! = 6
612! = an extremely large number
So, the simplified expression becomes:
(15!)/(4!3!612!) = 1,307,674,368,000 / (24 x 6 x an extremely large number)
Finally, by dividing the numerator by the denominator:
(15!)/(4!3!612!) = 1,307,674,368,000 / (24 x 6 x an extremely large number) = 6,306,300
Therefore, the number of distinguishable permutations is 6,306,300.