Misread the question

Find the total number of ways of formation of numbers, which are divisible by 3, utilizing 0,1,2,3,4 and 5 without repetition.

Well the question that i was solving asked to find out the number of ways of forming five digit numbers and since I had put a lot of effort in solving it just to find out that i had misread the question, I posted the question here, so as to verify whether my approach is correct or not. Btw I am a noob at combinatorics.

So what I did was as follows:-

1 digit numbers:- 1(i.e. the number 3)

2 digit numbers:-

Lets consider that the number is formed from two digits ${ x }_{ 1 }$ & ${ x }_{ 2 }$ and as the number should be a multiple of 3 so,

${ x }_{ 1 }+{ x }_{ 2 }=3n$ , where $n=1,2,3; 0{ <x }_{ 1 }\le 5; 0{ \le x }_{ 1 }\le 5$

Now, using multinomial expansion, we get $(x+{ x }^{ 2 }+{ x }^{ 3 }+{ x }^{ 4 }+{ x }^{ 5 })(1+x+{ x }^{ 2 }+{ x }^{ 3 }+{ x }^{ 4 }+{ x }^{ 5 })$ where we have to find the coefficient of ${ x }^{ 3n }, n=1,2,3$

And proceeding forward like this till the six digit numbers. Please do point out any other conditions that i might not have considered here and which would be required in further cases.

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Comments

@Rishabh Cool

Akhilesh Prasad - 5 years, 3 months ago

Just one point.. Shouldn't 0 be included since its divisible by 3 also...?

Rishabh Jain - 5 years, 3 months ago

Yeah, just forgot about that one, thanks any other thing that i might have missed.

@Akhilesh Prasad – I'm too a noob at Combinatorics ... So I cannot help ...Regrets for that....

@Sandeep Bhardwaj

Hint: application of divisibility rules, rule of product

Calvin Lin Staff - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$