# How many 3-digit numbers can be formed with the digits $1, 2, 3, 4$?

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?

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 :)

• So, my method was correct? – stonecraft bros Feb 21 at 6:09
• 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. – Aniket Gupta Feb 21 at 6:18

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....