Here are some more counting examples along with brief explanations on how to get the correct answers. Example 1: The English alphabet contains 21 consonants and 5 vowels. Consider strings of length 6 all of whose letters are different. a. How many such strings are there? Answer P(26, 6) = 26•25•24•23•22•21 b. How many of these contain exactly one vowel? There are 5 ways to choose the vowel, 6 places to put the vowel, P(21, 5) ways to place the consonants so the answer is the product of these numbers: 5•6•P(21, 5) c. How many contain exactly two vowels? This can be broken down into two parts—placing the vowels and placing the consonants. There are C(6,2) ways to choose the positions of the vowels, 5•4 ways to put two vowels into those positions, and finally P(21,4) ways to place the consonants. The final answer is the product of these three numbers. d. How many contain at least one vowel? Take the number of strings of length 6 and subtract the number of strings consisting only of consonants i.e. P(26, 6) – P(21, 6) Example 2: The English alphabet contains 21 consonants and 5 vowels. Consider strings of length 6 where letters may be repeated. a. How many such strings are there? Since there are 26 choices for each position in the string the answer is 266. b. How many of these contain exactly one vowel? There are 5 ways to choose the vowel, 6 places to put the vowel, and 21 5 ways to place the consonants for a total of 5•6•215. c. How many contain at least one vowel? Take the number of strings of length 6 and subtract the number of strings consisting only of consonants i.e. 266 - 216 Example 3: A committee of size 5 is chosen from a club with 20 males and 25 females. Find the number of committees in which a. a female is designated as the chair and the committee has 3 males and 2 females. C(20, 3)•C(25, 2)•2 (Choose the three females, choose the two females, and then select on of the two chosen women to be the chair.) Alternatively, choose the three males and then choose the woman who will be chair and then choose the other woman.C(20, 3)•25•24 b. One is designated as the chair and at least one is a male. First find the number of 5 member committees with no males and subtract that from the number of possible committees to get the number of 5 member committees with at least one male. Then, choose one of the five committee members to be chair giving: [C(45, 5) - C(25, 5)]•5 c. One is designated as the chair and exactly one male is on the committee. Choose the one male, choose four females, and then pick the chair from the five people selected to get: C(20, 1)•C(25, 4)•5 Example 4: More variations of the powerball problem a. How many powerball tickets are there in which no number is a multiple of 5? When the numbers on the white balls are considered, there are 55 – 11 = 44 balls whose numbers are not multiples of 5. Similarly, there are 42 – 8 = 34 numbers on the red balls that are not multiples of 5. Since the white balls can be chosen in C(44, 5) ways the total number of tickets satisfying this condition is C(44, 5)•34. b. How many ways are there to win a lesser prize? Let’s consider a few of the possibilities— for practice you can calculate the others. 1) How many tickets match only the powerball? Since the powerball must be matched, there is only one way this can be done. However, even though none of the white balls match, there must still be five white ball numbers chosen. This may be done in C(50, 5) ways. 2) How many tickets will match exactly three white balls and not the powerball? Of the five numbers chosen, you match exactly three which can be done in C(5, 3) ways. Since you must choose a total of five numbers and the other two do not match, there are C(50, 2) ways to choose the nonwinning numbers. There are 41 ways in which the powerball is not matched, so the number if tickets is C(5, 3)•C(50, 2)•41. 3) How many tickets will match exactly three white balls and the powerball. Using the same kind of reasoning as in the first two part, there is only one way to match the powerball and C(5, 3)•C(50, 2) ways to match exactly three white ball numbers so the answer is C(5, 3)•C(50, 2)• 1. 4) On how many tickets are all numbers chosen even? There are 27 even numbers on the white balls and 21 even-numbered red balls, so the answer is C(27, 5)•C(21, 1). Decision trees: Sometimes the only way to count something is to draw a decision tree in which each leaf represents a decision. (i.e. exhaustively find all of the possible ways to succeed.) For example, consider a best 3 of 5 tournament in which the teams A and B are evenly matched. We want to know in how many ways the tournament can be played. We’ll use the convention that if A wins we move to the left and if B wins we move to the right. It can be seen that in this case there are 20 ways for the tournament to be played.