Question of Third Kamikare
Third Kamikare
09/03/2017 at 22:30
Vũ Thị Hương Giang 09/03/2017 at 22:33
Numbers satisfying the problem condition are 5digit numbers abcde with 0 <a <b <c <d <e. We see that these numbers satisfy the following two properties:
1) The numbers a, b, c, d are different (the digits are different from 0)
2) The numbers are going up from left to right.
To count the numbers satisfying the two properties simultaneously, we perform the following two steps:
Step 1: Count the 5 digit number that satisfies property 1 (different digits) that temporarily omitted property 2.
We have:
 There are 9 ways to choose a number (from 1 to 9)
 After selecting a, there are 8 ways to choose the number b (from 1 to 9 but remove the selected digit).
 After selecting a, b, there are 7 ways to choose the number c (from 1 to 9 but remove the selected number a, b)
 After selecting a, b, c, there are 6 ways to choose d (from 1 to 9 but remove the selected a, b, c)
 After selecting a, b, c, d, there are 5 ways to choose e (from 1 to 9 but remove the selected a, b, c, d)
So there are all: 9 x 8 x 7 x 6 x 5 = 15120 numbers have 5 different digits and different digits 0.
Step 2: Calculate the number of numbers that satisfy the second characteristic (ie the numbers are going up from left to right).
We have commented: For any set of five digits [eg (1, 2, 3, 4, 5)] we can write 5 x 4 x 3 x 2 x 1 = 120 different numbers from five sets This number (similar argument above).
Of these 120 numbers, only one number satisfies the condition that the digits are going up from left to right.
Therefore the number of numbers satisfying the above property 2 is equal to 1/120 of the numbers satisfying condition 1.
And we have: the number of numbers satisfying simultaneously two properties 1 and 2 are: 15120: 120 = 126 numbers.
Third Kamikare selected this answer. 
FA FIFA Club World Cup 2018 16/01/2018 at 22:05
Numbers satisfying the problem condition are 5digit numbers abcde with 0 <a <b <c <d <e. We see that these numbers satisfy the following two properties:
1) The numbers a, b, c, d are different (the digits are different from 0)
2) The numbers are going up from left to right.
To count the numbers satisfying the two properties simultaneously, we perform the following two steps:
Step 1: Count the 5 digit number that satisfies property 1 (different digits) that temporarily omitted property 2.
We have:
 There are 9 ways to choose a number (from 1 to 9)
 After selecting a, there are 8 ways to choose the number b (from 1 to 9 but remove the selected digit).
 After selecting a, b, there are 7 ways to choose the number c (from 1 to 9 but remove the selected number a, b)
 After selecting a, b, c, there are 6 ways to choose d (from 1 to 9 but remove the selected a, b, c)
 After selecting a, b, c, d, there are 5 ways to choose e (from 1 to 9 but remove the selected a, b, c, d)
So there are all: 9 x 8 x 7 x 6 x 5 = 15120 numbers have 5 different digits and different digits 0.
Step 2: Calculate the number of numbers that satisfy the second characteristic (ie the numbers are going up from left to right).
We have commented: For any set of five digits [eg (1, 2, 3, 4, 5)] we can write 5 x 4 x 3 x 2 x 1 = 120 different numbers from five sets This number (similar argument above).
Of these 120 numbers, only one number satisfies the condition that the digits are going up from left to right.
Therefore the number of numbers satisfying the above property 2 is equal to 1/120 of the numbers satisfying condition 1.
And we have: the number of numbers satisfying simultaneously two properties 1 and 2 are: 15120: 120 = 126 numbers.
Numbers satisfying the problem condition are 5digit numbers abcde with 0 <a <b <c <d <e. We see that these numbers satisfy the following two properties:
1) The numbers a, b, c, d are different (the digits are different from 0)
2) The numbers are going up from left to right.
To count the numbers satisfying the two properties simultaneously, we perform the following two steps:
Step 1: Count the 5 digit number that satisfies property 1 (different digits) that temporarily omitted property 2.
We have:
 There are 9 ways to choose a number (from 1 to 9)
 After selecting a, there are 8 ways to choose the number b (from 1 to 9 but remove the selected digit).
 After selecting a, b, there are 7 ways to choose the number c (from 1 to 9 but remove the selected number a, b)
 After selecting a, b, c, there are 6 ways to choose d (from 1 to 9 but remove the selected a, b, c)
 After selecting a, b, c, d, there are 5 ways to choose e (from 1 to 9 but remove the selected a, b, c, d)
So there are all: 9 x 8 x 7 x 6 x 5 = 15120 numbers have 5 different digits and different digits 0.
Step 2: Calculate the number of numbers that satisfy the second characteristic (ie the numbers are going up from left to right).
We have commented: For any set of five digits [eg (1, 2, 3, 4, 5)] we can write 5 x 4 x 3 x 2 x 1 = 120 different numbers from five sets This number (similar argument above).
Of these 120 numbers, only one number satisfies the condition that the digits are going up from left to right.
Therefore the number of numbers satisfying the above property 2 is equal to 1/120 of the numbers satisfying condition 1.
And we have: the number of numbers satisfying simultaneously two properties 1 and 2 are: 15120: 120 = 126 numbers.

Trịnh Đức Phát 22/03/2017 at 12:50
Numbers satisfying the problem condition are 5digit numbers abcde with 0 <a <b <c <d <e. We see that these numbers satisfy the following two properties:
1) The numbers a, b, c, d are different (the digits are different from 0)
2) The numbers are going up from left to right.
To count the numbers satisfying the two properties simultaneously, we perform the following two steps:
Step 1: Count the 5 digit number that satisfies property 1 (different digits) that temporarily omitted property 2.
We have:
 There are 9 ways to choose a number (from 1 to 9)
 After selecting a, there are 8 ways to choose the number b (from 1 to 9 but remove the selected digit).
 After selecting a, b, there are 7 ways to choose the number c (from 1 to 9 but remove the selected number a, b)
 After selecting a, b, c, there are 6 ways to choose d (from 1 to 9 but remove the selected a, b, c)
 After selecting a, b, c, d, there are 5 ways to choose e (from 1 to 9 but remove the selected a, b, c, d)
So there are all: 9 x 8 x 7 x 6 x 5 = 15120 numbers have 5 different digits and different digits 0.
Step 2: Calculate the number of numbers that satisfy the second characteristic (ie the numbers are going up from left to right).
We have commented: For any set of five digits [eg (1, 2, 3, 4, 5)] we can write 5 x 4 x 3 x 2 x 1 = 120 different numbers from five sets This number (similar argument above).
Of these 120 numbers, only one number satisfies the condition that the digits are going up from left to right.
Therefore the number of numbers satisfying the above property 2 is equal to 1/120 of the numbers satisfying condition 1.
And we have: the number of numbers satisfying simultaneously two properties 1 and 2 are: 15120: 120 = 126 numbers.