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how many $3$ digit numbers can be formed by $1,2,3,4$, when the repetition of digits is allowed?

So basically, I attempted this question as-

There are 4 numbers and 3 places to put in the numbers: In the ones place, any 4 numbers can be put, so there are 4 choices in the ones place. Similarly for the tens and the hundreds place. So, the total choices are, by multiplication principle- $$4*4*4=64$$ And well and good, this was the answer.

But what if I reversed the method?

So I take some particular numbers, like $1,2,3$ and say that, well, $1$ can go in $3$ places, $2$ in $2$ places and $3$ in $1$ place, so by multiplication principle, there are $6$ ways of forming a $3$-digit number with $1,2,3$.

But there are $4$ different numbers. So the number of $3$-number combinations are- $(1,2,3)$,$(1,2,4)$,$(1,3,4)$,$(2,3,4)$. Each can be arranged in $6$ ways, so we get $24$ ways totally.

So why is my answer different here?

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Your answer is different because in the question, repitition is allowed, but you have only chosen combinations $(1,2,3),(1,2,4),(1,3,4),(2,3,4)$ in which there are no repeat numbers. So the combinations that you were supposed to include were $(1,1,1),(2,2,2),(3,3,3),(4,4,4), (1,2,2), (1,3,3),(1,4,4)...$ and so on.

Now, if you count the permutations for each of these combinations separately, and add it to 24, you will get 64.

Hope this helps :)

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  • $\begingroup$ So, my method was correct? $\endgroup$ – stonecraft bros Feb 21 at 6:09
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    $\begingroup$ Yes, your method would have been correct if you remembered to take combinations of repeated numbers. Unfortunately, it is a not easy to simply "know" when a method is correct. That requires practice and effort. Mathematics is all about practice. $\endgroup$ – Aniket Gupta Feb 21 at 6:18
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Well I think this is because when you do 4*4*4 , you consider that the digits in the number can repeat themselves. But in the second case , you take 3 cases in first place , 2 in second and 1 in third. So then you are considering numbers with all distinct digits..

Example- In first method- A number can be 111 But in second case - A number cannot be 111 , it will be like 123 or 124

That's why I think some numbers will not be counted....

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