jenyang5268 wrote:

Why is it that the answer to the following question is solved in the exact same way?

-How many different groups of 3 people can be formed from a group of 5 people?

Answer: 10

To solve this problem, don't I also use the equation: 5!/(3! x 2!) = 10 ? In this case, n!/k!(n-k)!.

Now I'm confused why in the previous question, there are 2U's and 3 R's, making a total of 5 letters. However, I am not selecting only 3 letters for my combination so why does is this problem solved with the formula n!/k!(n-k)! ? Please advise. Thanks!

The two questions you are asking are different.

-How many different groups of 3 people can be formed from a group of 5 people?

This is a combination since the order in which 3 people are selected is NOT important. It is just the selection. Hence 5C3 is correct.

Applying the formula, nCk = n! / (r! * n-r!), we have 5! / (2! * 3!) = 10.

Now coming to the original question, there are 5 paths where 2 paths are upwards(U) and 3 paths are sideways(S).

To reach from source to distination, the order of the path is important. So this means it is Permutation.

Let me take an example - How many ways can we arrange PAGES. 5 distinct letters are there and order is important so it is 5!.

How many ways can we arrange FEELS - Again 5 words with a E repeated twice. It will be 5! / 2!.

Similary coming to our problem, UUEEE can be arranged in 5! / (2! * 3!) since U is repeated twice and E is repeated thrice.

http://regentsprep.org/regents/math/permut/LpermRep.htmGoogle for Permutations with Repitions and you should find some information.