Approximations to n!, including Stirling's formula

The following are demonstrated: 1) ln(n!) = nlnn + O(n) 2) ln(n!) ~= nlnn - n 3) n! ~= sqrt(2*pi*n) * n^n * e^-n

How to calculate probabilities using x variables

A box contains 3n red balls and 3n white balls. 2n balls are selected at random without replacement, express the probability p that the balls selected contain n red and n white. Use stirlings formula to obtain the approximation p ~ root(3/(2*pi*n))

Math applications for business

• Linda received $875 on an investment of $14,000 at 15% annual simple interest rate. How long was the money invested? • Freddy borrowed $10,000 for a car for 3 years and paid $1,500 in simple interest. What was the rate of interest? • Diana owns municipal bonds valued at $14,500, which is 12% of her total investments. What is the value of her total investments? Round to the nearest dollar? • What is the straight-line depreciation for a machine that was purchased for $15,000, cost $750 to install, has a useful life of six years and a salvage value of $500. • Anita Tooms bought a new computer system. To pay for the system, she borrowed $2670 from the credit union at 7 1/3% interes... click for more

How many ways are there to paint the 10 identical rooms in a hotel; Find the exponential generating function...

(See attached file for full problem description with equations) --- Exercise #1 A) How many ways are there to paint the 10 identical rooms in a hotel with five colors if at most three rooms can be painted green, at most three painted blue, at most three red, and no constraint on the other two colors, black and white. B) Show that is the generating function for the number of ways a sum of r can occur if a die is rolled any number of times. Exercise#2 A) Find the exponential generating function for the number of ways to distribute r distinct objects into n distinct boxes with no empty box. Consider n a fixed constant. B) Determine is called a Stirling number of the second ... click for more

Evaluate the sum..

Evaluate the sum (using generating functions) A) 0+3+12+…+3n2. B) 4x3x2x1+5x4x3x2+…+n(n-1)(n-2)(n-3)