0
$\begingroup$

For example, how would I calculate how many $5$ digit numbers I can write using only digits $0, 2, 2, 3, 3$?

Is $4\times5\times5\times5\times5$ correct or am I missing something?

$\endgroup$
  • $\begingroup$ Does for example 02233 count as a 5 digit number? Or is that a four digit number? $\endgroup$ – Harald Hanche-Olsen Jan 7 '14 at 22:50
  • $\begingroup$ So each number you create has to use 2 2's, 2 3's, and 1 0? $\endgroup$ – Mike Jan 7 '14 at 22:51
  • $\begingroup$ Nope, I don't think 02233 is a number. $\endgroup$ – Tool Jan 7 '14 at 22:52
  • $\begingroup$ What about 33333? I assume no, but it would be less clear if those digits were all different. $\endgroup$ – Mike Jan 7 '14 at 22:59
5
$\begingroup$

Total non similar combinations possible=$\dfrac{5!}{2!\cdot 2!}=30$

Number of combinations where digit "$0$" is at start(assuming $02233$ isn't a 5-digit number for instance)=$\dfrac{4!}{2!\cdot 2!}=6$

Number of valid 5-digit numbers=$30-6=24$

Note: If "$0$" is allowed at the start of a number, the answer is $30$.

$\endgroup$
2
$\begingroup$

Since 0 isn't allowed as a leading digit, you have 4 choices for where to put it, and then you have $\binom{4}{2}=6$ choices for where to place the 2's;

so there are $4\cdot6=24$ possible 5-digit numbers.

$\endgroup$
2
$\begingroup$

The reason that 4×5×5×5×5 is the wrong answer is because it will have duplicates in it. How? Here's how

Let's assume we take 2 0 2 3 3 as first number, now if I change the first and second 2 I still get the same number, i.e. 2 0 2 3 3. Hence with your answer, you are considering all these numbers as the different ones. To get the correct answer, K Rmth's answer is good enough.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.