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How many different 7-digit numbers there are with exactly two "8" and three "4"?

It is worth noticing that a number can not start with "zero", so:

0044488 is not a possible 7-digit number. This is a combinatorics problem, so we are talking about natural numbers.

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Suppose your number begins with $8$, then you have to fill in the remaining six places with $4,4,4$ and a $8$ and two digits $a,b$, where $a,b \in \{0,1,2,3,5,6,7,9\}$. So for that:

  • we choose three places for $4'$s in $\binom{6}{3}$ ways.
  • we choose a place for the $8$ in $\binom{3}{1}$ ways.
  • with four places gone, we choose a number for the remaining fifth place in $8$ ways (from the set listed above).
  • likewise the last remaining sixth place can also be filled in $8$ ways.

Thus the total number of seven digit numbers starting with $8$ and having a total of exactly three $4'$s and two $8'$s is $$\binom{6}{3}\binom{3}{1}\cdot 8^2$$

Likewise the total number of seven digit numbers starting with $4$ and having a total of exactly three $4'$s and two $8'$s is $$\binom{6}{2}\binom{4}{2}\cdot 8^2$$

The last case is when the number begins with one of the digits in $\{1,2,3,5,6,7,9\}$, then the first place can be filled in $7$ ways and using the same reasoning as above, the number of seven digit nos. satisfying the property are $$7 \cdot \binom{6}{3}\binom{3}{2}\cdot 8$$

Hopefully you can take it from here.

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  • $\begingroup$ So the answer would be the sum of this numbers, correct? $\endgroup$ – user3347814 Aug 22 at 3:31
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    $\begingroup$ @user3347814 The answer is yes BUT I would strongly recommend you to brush up the basics first so that such doubts will not arise. $\endgroup$ – Anurag A Aug 22 at 5:28
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Here's an outline:

Assume the first digit is neither 8 nor 4.

How many choices are there for the first digit?

Once you pick that, how many options do you have to place the two 8's in the remaining spots?

Then, how many options do you have to place the three 4's?

And finally, what are the options for the remaining spots?

Next, do the same, assuming the first digit is 8. And the same if it starts with 4. Finally, add up the three counts you got.

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  • $\begingroup$ Could you please provide a result? $\endgroup$ – user3347814 Aug 22 at 2:37
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    $\begingroup$ What part don't you understand? $\endgroup$ – TorsionSquid Aug 22 at 2:38
  • $\begingroup$ Everything, basically. $\endgroup$ – user3347814 Aug 22 at 2:40
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    $\begingroup$ Well in that case, providing the result isn't going to help you understand... $\endgroup$ – TorsionSquid Aug 22 at 2:41

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