23 FACTORIALSBY THE SYMBOL n! ("n factorial") we mean the product of consecutive numbers 1 through n.
The order of the factors does not matter, whether backwards or forwards. 0! is defined as 1. 0! = 1. (The usefulness of this definition will become clear as we continue.) Example. 6! = 1· 2· 3· 4· 5· 6 = 720 A convenient way to calculate this is to wait to multiply 2· 5 = 10. Then 3· 4· 6 = 12· 6 = 72 -- times 10 is 720. Calculations with factorials are based on this fact: Any factorial less than n! is a factor of n!. Example 1. 6! is a factor of 10!. For,
Solution. 6! is a factor of 8!, so it will cancel leaving 7· 8 in the
Problem 1. What factorial is each of these? (Consider what each symbol means.) To see the answer, pass your mouse over the colored area. a) 3!· 4 = 4! For, 3!· 4 = 1· 2· 3· 4 = 4!. b) 6!· 7· 8 = 8! c) (n − 1)! n = n! d) (n − k − 1)! (n − k) = (n − k)!
Solution. If we multiply both terms on the left by n, then
Now, (n − 1)! is the product up to the number just before n. Therefore, (n − 1)! times n itself is n!.
Example 5. Show: n! k + n! (n − k + 1) = (n + 1)! Solution. On the left-hand side, n! is a common factor:
(n + 1) is the number after n. Problem 4. Write out the steps that show how to transform the left-hand side into the right-hand side. Again, consider what each symbol means.
Multiply both terms on the left by 4:
Multiply both terms on the left by k:
Since 12! = 11!· 12, divide both terms on the left by 12:
Since (n − k)! = (n − k − 1)!· (n − k), divide both terms on the left by (n − k):
e) 5!· 4 + 5!· 2 = 6! 5! is a common factor: 5!· 4 + 5!· 2 = 5!(4 + 2) = 5!· 6 = 6! Next Topic: Permutations and combinations Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |