Important! These are all exam-level problems. Do not attempt these problems without a solid foundation in the subject and use them for exam practice.
1. How many anagrams are there of the word CHOCOLATE? An anagram is a word made up of the letters of original word in any order.
First, it is important to note that the word has 10 letters, with 2 C's and 2 O's. So, we have 10! ways of rearranging the words, but we must keep the 1 C's and 2 O's as if they are the same thing. Hence, the final answer is 10! / 2!2!. 2. How many solutions does x + y + z = 10 have if all variables must be positive integers?
pd+1-k. It's important to think of this as a stars and bars problem. We have 10 stars, and 3 groups where each group must have atleast one star, as in each variable must be atleast 1 (positive integers). Then, the real problem becomes putting 7 stars into 3 groups, which is n+k-1 choose k-1, which is 9C2. 3. In poker, where each hand has 6 cards, how many ways are there to pick 3 distinct pairs?
Pick 3 distinct numbers from the possible 13, and then for each pair, choose 2 suits from a possible 4. We get 13 choose 3 * (4 choose 2)3.