Combinations and Probability help please! Thank you! Problem 1: Combinations. Re
ID: 2932874 • Letter: C
Question
Combinations and Probability help please! Thank you!
Problem 1: Combinations. Recall that Tn! is the number of different ways to pick k objects out of n different objects without considering the order nor replacement. 1. Use (1) to show that Tt n-k TL 2. Next we show (2) by Probability. Consider the following question: if you have n friends, and you pick a group of k friends out of these n people to attend your birthday party. How many different groups you can pick? 3. Consider a relevant but different question: for the same n friends of yours, pick a group of n -k friends NOT to attend your birthday party (everyone who isn't picked will attend). How many different groups can you pick? 4. Combine the two questions above to prove (2).Explanation / Answer
1. nC(n-k) = n! / [(n-(n-k))! (n-k)!]
= n! / [k! (n-k)!]
= nCk.
2. Since there are n friends and the party needs k friends, this can be done in nCk ways.
3. Since there are n friends and n-k friends are not in the party, we can choose in nC(n-k) ways.
4. If n-k friends are not in the party, then the remaining k friends must be in the party. Conversely if k friends are in the party, then n-k friends are not in the party.
Therefore 2 and 3 count the same thing.
=> nCk = nC(n-k).