Permutations and combinations

Some questions I'm having problems with:

1. Find the number of ways of seating 10 people around a table if two people must not sit together.
i understand the number of seating arrangements for 10 people around a table is 9! I'm not sure how to calculate 2 people not sitting together.
2. The number of possible different PINs with a combination of 4 numbers and 2 letters is.
I thought possibly 10!/(10-4)! X 24!/(24-2)! Would yield the answer but it didn't....
Thanks for any help

*edit* sorry for not posting my working, I've edited it in

Lemonmelon said:

Some questions I'm having problems with:

1. Find the number of ways of seating 10 people around a table if two people must not sit togeather
2. The number of possible different PINs with a combination of 4 numbers and 2 letters is

Click to expand...

What are your thoughts?

Please share your work with us ...even if you know it is wrong

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Lemonmelon said:

1. Find the number of ways of seating 10 people around a table if two people must not sit together

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What formula or method have they given you for working with orderings in circles? Assuming you set the two people as one unit, so you've got nine units (eight people, plus the one pair), in how many ways can you order the units? In how many ways can you order all ten people, as individuals? Since the seatings using the pair as one unit are the seatings that you do not want, how many seatings are there which are of the type you want?

Lemonmelon said:

2. The number of possible different PINs with a combination of 4 numbers and 2 letters is

Click to expand...

How many digits are there? (Hint: Count the digits between 0 and 9, inclusive.)

How many letters are there? (Hint: Twenty-six.)

In how many ways can you fill each of the number slots? In how many ways can you fill each of the letter slots?

In how many ways can you order the six different slots?

Please reply showing all of your thoughts and efforts so far. Thank you!

Lemonmelon said:

1. Find the number of ways of seating 10 people around a table if two people must not sit together.
i understand the number of seating arrangements for 10 people around a table is 9! I'm not sure how to calculate 2 people not sitting together.

2. The number of possible different PINs with a combination of 4 numbers and 2 letters is.
I thought possibly 10!/(10-4)! X 24!/(24-2)! Would yield the answer but it didn't....

Click to expand...

For the first problem, let's number the guests from 1 to 10. We'll say that guests 1 and 2 absolutely can't sit next to each other, no matter what. So, go through the possible arrangements. Because the guests are sitting around a table, guest 1 can sit anywhere, and it won't matter. Guest 2 now has 9 seats left open to pick from. But guests 1 and 2 can't sit next to each other. So, how many seats does guest 2 really have to choose from? Then when guest 3 comes, how many chairs are there to pick from? And so on until the final guest.

For the second problem, there are two problems I see. One is in the numbers part, you're using a combination formula that doesn't allow for repeated digits. Unless the problem specifically stated otherwise, repeated digits are allowed. For instance, 1111 is a valid combination, but you're not counting that possibility with your formula. Instead, approach it from the bottom up. For the first digit of the PIN, how many possibilities are there? For the second digit? Then how many total possibilities are there for just the digits?

Then on the letters portion, there are two errors. Again, you're trying to use a combination formula with no repeated letters. Also, even if that was the right formula to use, you're saying there are 24 letters. Is that right? But anyway, if we work with it the same as the numbers part... How many possibilities are there for the first letter? For the second letter? To arrange both letters? Now you know how many number combinations there are and how many letter combinations there are, so how many total PINs are there?