How many $4$-digit lock combinations are possible if each digit in the code may appear at most twice?

Question

Question A lock combination code is made up of $4$ numbers ($0-9$). Each number can occur at most twice, e.g. $4764$ would be allowed but not $4464$ as the number $4$ has occurred more than $2$ times.Therefore, how many possible combination codes are there?

I know that there are $10,000$ possible combinations if repetitions are allowed. However I'm unsure as to how to answer the question. Any help is greatly appreciated, thanks!

Answer 1

Here is a possible solution using the Principle of Inclusion/Exclusion:

$\textbf{Case 1}$ (digit is repeated $3$ times):

Choose the repeated digit in $10 \choose 1$ ways.

Choose the remaining digit in $9 \choose 1$ ways.

Order the digits in $\frac{4!}{3!} = 4$ ways.

$\textbf{Case 2}$ (digit is repeated $4$ times):

Choose the repeated digit in $10 \choose 1$ ways.

Order the digits in $\frac{4!}{4!} = 1$ way.

So the number of combinations that don't satisfy your constraint is $(10 * 9 * 4) + (10 * 1) = 370$. Subtracting these from the $10,000$ total combinations yields $\textbf{9,630}$ ways.

Answer 2

Solution without using the Principle of Inclusion/Exclusion:

Add by number of pairs of numbers: $0$ pairs $+$ $1$ pair $+$ $2$ pairs

$=$ (pick $4$ of the $10$ order matters) $+$ (pick which will be repeated and where and the other $2$ numbers order matters) $+$ (pick number for each pair and location of first and second pair)

$=P(10,4) + 10*{{4}\choose{2}}*P(9,2) + {{10}\choose{2}}*{{4}\choose{2}}*1$

$=5040$ $+$ $4320$ $+$ $290$

$= 9630$