Differences between Permutations and Combinations I am a little confused by these questions, I have a very basic understanding of them but I am hoping that someone might be able to explain them to me a little better.
There are ten different types of pens (example: 1 with blue ink, 1 with black ink etc.) that a company offers in their production line. This is true for all the questions below:


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*You need to order all your pens for each day of the week on one single day (and have them delivered on that day). It does not mater what type of pens you get, but you need to make sure you have one for every day of the week. How many orders are now possible?


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*I am assuming this would be something like 10 choose 7.


*The pen company offers you a special with 2 blue inked pens, 3 red inked pens, and 2 black inked pens which are to be delivered one per day for the entire week. How many orders are possible if you need to tell the company what pens you want and on what day. 

*You need to order 7 pens on one single day, and have them delivered on that day. You now need a different pen for each day of the week How many orders are possible?
Thank you for any help ahead of time!
 A: It actually depends. If the pens are indifferent, use combinations. If not, use permutations. Some problems, like the second one, use a combination of both. Good luck! Ask me if you need more help.
A: For number $1,$ you need to specify seven not necessarily different pen types taken from a set of $10$ possible types.  Arrangement doesn't matter since the pens are all to be delivered on one day.  If the pen types are $A,B,C,D,E,F,G,H,I,J$ then examples of valid pen orders would be sequences like $BCDEFGH$ or $AAAAAAA$ or $AACCEEG.$  You can use the stars and bars technique to count these.  The "bars" are separator marks between different types.  So the sequence $BCDEFGH$ could be written $\mid B\mid C\mid D\mid E\mid F\mid G\mid H\mid\mid$.  You can think of the bars as divisions between different "bins", one bin for each type of pen.  In this example, the $A$ bin is empty, as are the $I$ and $J$ bins.  The sequence $AACCEEG$ could be written $AA\mid\mid CC\mid\mid EE\mid\mid G\mid\mid\mid$.  Since a sequence is specified by saying how many items each bin contains, the letters can all be replaced with a single symbol,  say a star.  So the first example would be represented by $\mid *\mid *\mid *\mid *\mid *\mid *\mid *\mid\mid,$ which indicates that the $B,C,D,E,F,G,$ and $H$ bins each contain one item, while the second example would be represented by $**\mid\mid **\mid\mid **\mid\mid *\mid\mid\mid,$ which indicates that the $A,C,$ and $E$ bins each contain two items and the $G$ bin contains one item.  Observe that since there are $10$ bins, there will always be nine bars.  Since there are seven items, there will always be seven stars.  So it remains to enumerate the number of sequences with nine bars and seven stars, which can be done using combinations.
For number $2,$ the numbers of pens of each color to be delivered are fixed.  The only thing that needs to be specified are the days on which blue pens are to be delivered, the days on which red pens are to be delivered, and the days on which black pens are to be delivered.  You can think of this as a process of choosing days of the week: first pick two days for delivery of blue pens, then pick three from the remaining days for the delivery of red pens, and finally pick two from the remaining days for the delivery of black pens.  Alternatively, using $B$ for blue, $R$ for red, and $K$ for black, you need to find all arrangements of the sequence $BBRRRKK.$  The answer to this is a multinomial coefficient.
Number $3$ is more straightforward: since the pens are to be delivered on one day, you are merely choosing seven different types from a set of $10$ possibilities. 
