Each of the following is a single combination, permutation or exponential. Find the number of ways to:
1. Choose any 9 letters out of the 26-letter alphabet.
2. Make a 9-letter code from the 26-letter alphabet.
3. Choose any 7 of the 10 digits, 0, 1, 2, ..., 9.
4. Make a 7-digit phone number.
5. Choose a president, vice president, secretary and treasurer from a club of 50 people.
6. Select 4 people from 50 to work together on a project.
7. How many ways can 7 randomly chosen people in a group of 30 be lined up for a photo?
Each of the following is a product of combinations, permutations, and or exponentials.
Break the task into steps, count each step and use the multiplication principle.
8. i) How many 10-long codes of letters of the 26 letter alphabet have exactly two a's, exactly 3 b's and
no other repeats?
ii) How many 10-long codes of letters of the 26 letter alphabet have exactly two a's, exactly 3 b's and
any other letters, other repeats allowed?
9. How many full houses consisting of 3 of a kind and one pair can be chosen from a standard 52-card
deck?
10. How many 5-card poker hands have two pair.
11. A foreign language club has 20 German speakers, 35 French speakers and 18 Spanish speakers.
How many ways can one choose a representative group consisting of 5 German, 7 French and 10
Spanish speaking people?
12. A multiple choice test has 15 questions with 5 answer choices for each. How many ways could a
person answer the whole test and answer exactly two questions incorrectly?
Each of the following involves all the above and possibly an addition rule or complement rule.
13. How many ways can the foreign language club of #11 choose 4 or 5 German speakers, 3 or 4 French
speakers and 6 or 7 Spanish speakers?
14. i) How many 7-letter codes made from the 26 letter alphabet contain no a's?
ii) How many such codes contain at least one a?
15. Two six sided dice are tossed. In how many outcomes is the sum of the top numbers:
i) less than or equal to 8?
ii) more than 4?
16. How many 5-card poker hands contain all hearts or a straight/straight flush ?
A straight is all cards in sequence. A straight flush is all cards in sequence and all the same suit.