Problem: Find the number x if the ratio of the factorial of x to the square of the factorial of another number which when increased by 50% gives the required number is 1.25.

Solution:
Let's break down the problem into steps:

Step 1: Understand the problem
We are given a ratio of two factorials and we need to find one of the numbers. The problem involves understanding and using the concept of factorials.

Step 2: Use the formula for factorial
The formula for factorial of a number n is:
n! = n x (n-1) x (n-2) x ... x 2 x 1

Step 3: Write the equation given in the problem
We are given that the ratio of the factorial of a number x to the square of the factorial of another number (let's call it y) which when increased by 50% gives the required number is 1.25. This can be written as:
x! / y!^2 = 1.25
Also, we know that y + 0.5y = x, which can be simplified as 1.5y = x.

Step 4: Rewrite the equation in terms of y
Using the relationship between x and y, we can rewrite the equation as:
x! / y!^2 = 1.25
(1.5y)! / y!^2 = 1.25

Step 5: Simplify the equation
We can simplify the equation by expanding the factorials:
(1.5y)(1.5y - 1)(1.5y - 2)...(y+1) / y!y! = 1.25

Step 6: Cancel out the common terms
We can cancel out the common terms from the numerator and denominator:
(1.5y)(1.5y - 1)(1.5y - 2)...(y+1) / y!y! = 1.25
(1.5y)(1.5y - 1)(1.5y - 2)...(y+1) = 1.25y!^2

Step 7: Solve for y
We can solve for y by simplifying the left-hand side and equating it to the right-hand side:
(1.5y)(1.5y - 1)(1.5y - 2)...(y+1) = 1.25y!^2
1.5^y (y!) (1.5^4-1)(1.5^3-1)...(1.5^2-1) = 1.25y!^2
1.5^4y (y!) (4/3)(2/3) = 1.25y!^2
y = 6

Step 8: Find x
Using the relationship between x and y, we can find x:
x = 1.5y
x = 1.5(6)
x = 9

Therefore, the answer is (c) 9.