Answer to Question #4739 in Combinatorics | Number Theory for Hussain Mirza

Question

How many ordered 5-subsets of {0,...,9} are there? How many of these start with 0? How many positive integers of exactly 5 digits, but with no two digits the same are there? (In base 10)

Answer 1

Let \{a,b,c,d,e\}
be any ordered 5-subset of {0,...,9}.

> How many ordered 5-subsets of
{0,...,9} are there?
Thebn the first number a can be choosen by 10 ways, the
next number b can be choosen by 9 ways, and so on.
Therefore the total number
of ordered 5-subset of {0,...,9} is equal to
10*9*8*7*6 = 30240

How
many of these start with 0?
If we choose a=0, then the next number b can be
choosen by 9 ways from the set {1,...,9}, the next number c can be
choosen by
8 ways and so on, therefore the total number of ordered 5-subset of {0,...,9}
started from 0 is equal to
9*8*7*6 = 3024

> How many positive
integers of exactly 5 digits, but with no two digits the same are there?
Each
positive integers of exactly 5 digits starts with 1,...,9, therefore the total
number of such integers is
9*9*8*7*6=27216