Timothy's 9-digit Dilemma

Probability Level 3

Timothy had finished writing all 9-digit numbers that contains numbers (1-9) and without repeating the same digit in the same number, and he was rather pleased with himself.

'There all 362880 numbers, and it took me just 2 days!', he boasted to his friend.

'Why would you torment yourself like that?', ask the very confused friend.

'It's factorial fun!' came the reply

At that very moment, the enthusiastic maths teacher suddenly appeared! 'Aha! I see what you have there! When I was your age I did the exactly same thing! But do you know how many of these numbers are divisible by 6?'

The friend, feeling very awkward, decide to fade away into the background.

'Well... I know that all numbers divisible by 6 are divisible by 2 AND 3; but would I need to divide every individual number with 6 over and over?', stated Timothy feeling glum.

The teacher replied, ' Oh no, you do not need to, Just figure out what has to be done; I know you learnt factorials.'

Timothy managed to find the right answer, and his teacher commended him. What was Timothy's answer?

The answer is 161280.

5 solutions

Johanan Paul
Apr 6, 2014

Using the divisible rule of 3, we can conclude that all numbers that contains the digits: 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition are always divisible by 3.

1+2+3+4+5+6+7+8+9=45
4+5 = 9 9/3 = 3

So, based on that information, we can rephrase the question as: How many 9-digit numbers that contain only the numbers 1-9 are divisible by 2?

The 4 even numbers from 1-9 are: 2, 4, 6, 8

Since all even numbers are divisible by 2, we have to sift through all the numbers that end with an even number. In other words the first 8 digits can be the rest of the 8 number while the last has to be an even number.

To find out the number of ways 8 things can be arranged is by the fractorial 8! (which equals 40320) We then take this number and multiply it by 4 (the number of even numbers in 1-9)

We can then get the answer 4 * 8! = 4 * 40320 = 161280

Brilliant :)

Vishwa Shah - 7 years, 1 month ago

Adrian Neacșu
Apr 20, 2014

For the numbers to be divisible my 6 they should have the digit sum divisible by 3 (which is always true) and the number should be even.

There are 8! numbers that end with 2.

There are 8! numbers that end with 4.

There are 8! numbers that end with 6.

There are 8! numbers that end with 8.

There are 4x8! numbers divisible by 6.

Ron Balter
Jan 12, 2016

OK so I looked at this not exectly as math but, since the sum is 45 it's always divisible by 3. And in order for it to divide by 2 its the arrangement when (2,4,6,8) are the last digit, So the number of numbers ending with even digit is 4/9 of all the options, hence 4/9 * 362880 = 161280 Which by the way 362880=9! And 4/9 * 9! = 4*8! as all the above answers

Shamik Basu
Apr 27, 2014

since the nos are of consecutive digits from 1 to 9 they are divisible by 9. so they are also divisible by 3. so we need to find only the no of numbers divisible by 2 i.e. 4*8! which is the required answer.

Murlidhar Sharma
Apr 13, 2014

The numbers divisible by 2 were divisible by 6, as sum of 1 to 9 is divisible by 3. So the answer would be (362880/9)*4=1612832

your answer is wrong . . . .

all numbers of 9 digit in given question is divisible by 3.....no doubt

only ans:8!*4=161280

Ashutosh Kumar - 7 years, 1 month ago

Timothy's 9-digit Dilemma

The answer is 161280.

5 solutions

0 pending reports