10. Combinatorics & Probability

Evaluate the expression.

$\frac{12}{4!}$

Evaluate the expression.

$\frac{9!}{7!}$

Evaluate the expression.

$\frac{16!}{12!\cdot 4!}$

Write the first 4 terms of the sequence $a_n=n^2\cdot (n-1)!$

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=(n+1)!/n^2

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=−2(n−1)!

In Exercises 23–28, evaluate each factorial expression. 18!/16!

In Exercises 23–28, evaluate each factorial expression. 20!/2!18!

In Exercises 23–28, evaluate each factorial expression. (2n+1)!/(2n)!

The factorial of a positive integer n can be computed as a product. n! = 1 * 2 * 3 *. . . * n

Calculators and computers can evaluate factorials very quickly. Before the days of modern technology, mathematicians developed Stirling’s formula for approximating large factorials. The formula involves the irrational numbers p and e.

n! = √2πn * n^n * e^−n

As an example, the exact value of 5! is 120, and Stirling’s formula gives the approximation as 118.019168 with a graphing calculator. This is “off” by less than 2, an error of only 1.65%. Work Exercises 59–62 in order. Use a calculator to find the exact value of 10! and its approximation, using Stirling’s

formula.