Permtations and Combinations: hockey players

1) A hockey team consists of 19 players. In how many ways can:

a) a team captain and the assistant captain be chosen?

My answer \(\displaystyle \L\\\frac{{19!}}{{17!}} = 342\) or

\(\displaystyle \L\\{}_{19}P_2 = 342\)

b) Three union representatives be chosen?

My answer using same notation as above. 5814

2) How many different "words" can be made using all the letters of the word SUCCESS ?
My answer

\(\displaystyle \L\\\frac{{7!}}{{3!2!}}\)

420

I'm feeling pretty good about these questions. Please let me know if I am missing out something...(terrible feeling thinking I am missing something)
Thank you
Louise

Remember that the order in which team captain and the assistant captain, or the three union representatives are selected is not important, only the fact that they are selected. Therefore, you need to use the combination formula (If that's what it's called).

The second question is a little bit more complicated, since you need to take care not to double count words when they are identical. Give me a little more time. You do need to use permutations though as the order is important.

Louise Johnson, you have #2 correct and #1(a) is also correct.

But #1(b) should be \(\displaystyle \frac{{19!}}{{\left( {3!} \right)\left( {16!} \right)}}.\)

Note that is \(\displaystyle \frac{{5814}}{{\left( {3!} \right)}}.\)

DOH!! I should have seen that three!! Thank you I was getting worried I had everything incorrect.
Sincerly
Louise

Are you sure you only need the combination of 7 letter words? If so, then your answer is correct.

Dear Trebor, The question asks: How many different "words" can be made using all the letters of the word SUCCESS?

By words they just mean any arrongement of letters and doesn't need to actually make a proper english word.
By reading the "can be made using all the letters" sounds to me like it should be combinations of 7 letter "words"

Am I reading this question incorrectly?
Thank you
Louise

Yes, they mean arrangements of the word SUCCESS. You are correct.

\(\displaystyle \frac{7!}{3!2!}=420\)

WHEWWW!! Thank you for that confirmation.. Boy I was getting worried that I was reading all my questions wrong.
Thanks again
Louise