A problem of combinatorics — The number of 5 digit numbers of different digits with...

  • Thread starter sahilmm15
  • Start date
  • #1
100
27
Homework Statement:
The total number of 5 digit numbers of different digits in which the digit in the middle is the largest is ?
Relevant Equations:
N/A
I am scratching my head on the problem but cannot figure out what to do. I tried in the following way:-
For (9)
_ _ _ _ _ . The middle place is fixed means only 1 way there. For the first place 9 ways (excluding 0) , second place (9 ways again because 9 digits are left excluding 9 and 0). Third place is the middle one which is already filled, fourth place 8 ways and last place 7 ways .

So total no of digits in which the digit in the middle is largest ( in this case it is 9 ) are 9*9*1*8*7. The same method would go for other numbers. Is my method correct or am I approaching the problem incorrectly?? I cannot find the answer.
 

Answers and Replies

  • #2
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
17,164
8,963
Let's see a list of such numbers.
 
  • #3
100
27
After commenting on my method, Can you explain me why author has subtracted few terms in his method in the image below.
 

Attachments

  • IMG_20201225_203247.jpg
    IMG_20201225_203247.jpg
    127 KB · Views: 32
  • #4
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
17,164
8,963
After commenting on my method, Can you explain me why author has subtracted few terms in his method in the image below.
A five-digit number that starts with ##0## is only a four-digit number and doesn't count.
 
  • #5
1,961
1,209
A five-digit number that starts with ##0## is only a four-digit number and doesn't count.
I would interpret '5 digit numbers' to mean 'unique strings of length 5 each position of which is a decimal numeral', so that, with the 'middle digit is greatest' and 'digits are all different' constraints, the normally-lowest-sorting possibility would be 01423 and the normally-highest-sorting possibility would be 87965.
 
  • #6
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
17,164
8,963
I would interpret '5 digit numbers' to mean 'unique strings of length 5 each position of which is a decimal numeral', so that, with the 'middle digit is greatest' and 'digits are all different' constraints, the normally-lowest-sorting possibility would be 01423 and the normally-highest-sorting possibility would be 87965.
A number is a number and a string is a string. 01423 is a five-digit string but a four-digit number. In the same way that $$0x^2 + x + 1$$ is not a quadratic expression.
 
  • Like
Likes Vanadium 50 and sysprog
  • #7
1,961
1,209
A number is a number and a string is a string. 01423 is a five-digit string but a four-digit number. In the same way that $$0x^2 + x + 1$$ is not a quadratic expression.
I was thinking of pushbutton locks that have buttons that are numerally marked 0 through 9, which locks are such that until a reset button is pressed, it doesn't matter in which order or how many times the numerally-marked buttons are pressed ##\dots##
 
  • #8
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,229
6,838
... of different digits ...

For (9)
_ _ _ _ _ . The middle place is fixed means only 1 way there. For the first place 9 ways
If the middle is 9, no others can be 9.
 
  • #9
100
27
If the middle is 9, no others can be 9.
Hmm 8 ways.
 

Related Threads on A problem of combinatorics — The number of 5 digit numbers of different digits with...

Replies
1
Views
2K
Replies
0
Views
1K
Replies
5
Views
3K
Replies
3
Views
7K
  • Last Post
Replies
7
Views
1K
Replies
5
Views
4K
Replies
0
Views
2K
Top