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

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

PeroK
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Let's see a list of such numbers.

After commenting on my method, Can you explain me why author has subtracted few terms in his method in the image below.

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PeroK
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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.

• sahilmm15
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.

PeroK
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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.

• Vanadium 50 and sysprog
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##

haruspex
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... 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.

If the middle is 9, no others can be 9.
Hmm 8 ways.