Understanding the problem:
The given equation is np3 = 7.(n-1p3), where n is a variable that we need to determine. To solve this equation, we need to understand the concept of permutations and how they relate to the expression.

Permutations:
Permutations are a way of arranging objects in a specific order. The formula for finding the number of permutations of n objects taken r at a time is given by nPr = n! / (n-r)!, where n! represents the factorial of n.

Simplifying the equation:
Let's simplify the equation np3 = 7.(n-1p3) step by step.

1. Start with the left side of the equation: np3
- This represents the number of permutations of n objects taken 3 at a time.
- By using the formula for permutations, we can express it as n! / (n-3)!.

2. Move to the right side of the equation: 7.(n-1p3)
- This represents the number of permutations of (n-1) objects taken 3 at a time, multiplied by 7.
- By using the formula for permutations, we can express it as 7 * ((n-1)! / ((n-1)-3)!).

3. Now, we can rewrite the equation as follows:
n! / (n-3)! = 7 * ((n-1)! / ((n-1)-3)!)

Canceling out terms:
To further simplify the equation, we can cancel out common terms on both sides.

1. Cancel out the factorials:
- The terms n! and (n-1)! can be canceled out on both sides.

2. Cancel out the (n-3)! terms:
- We can cancel out (n-3)! terms on both sides.

After canceling out the common terms, the equation becomes:

n * (n-1) * (n-2) = 7 * (n-1) * (n-2)

Determining the value of n:
Since we have canceled out the common terms, we can now determine the value of n.

1. Divide both sides of the equation by (n-1) * (n-2):
- This will eliminate the terms (n-1) and (n-2) from both sides.

After dividing, the equation becomes:

n = 7

Therefore, the value of n is 7.

Conclusion:
By simplifying the given equation and canceling out common terms, we determined that the value of n is 7.