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

Question #4739

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)

Expert's answer

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

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

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