counting techniques exercises

2. Counting Techniques
20. In how many ways can 9 people be seated in a round table?
The Fundamental Counting Principle
For a sequence of events in which the first event can occur n1 ways, the second in n2 ways, the third in n3 ways, and so on until the kth event,
then events together can occur a total of 𝑛1 ∙ 𝑛2 ∙ 𝑛3 ∙ ⋯ ∙ 𝑛 𝑘 ways.
The Factorial Notation
The factorial of a positive integer n, denoted by n! (read as “n factorial”) is the product of decreasing positive whole numbers. In symbols,
𝑛! = 𝑛 ∙ (𝑛 − 1) ∙ (𝑛 − 2) ∙ ⋯ ∙ 3 ∙ 2 ∙ 1. By special definition, 0! = 1.
Permutation: an ordered arrangement of objects Combination:
𝑛!
a. When items are all different : ⬚ 𝑛 𝑃𝑟 = arrangement of objects without regard to order
(𝑛−𝑟)!
𝑛!
b. When some items are identical: 𝑛!
⬚ 𝑛 𝐶 𝑟 = (𝑛−𝑟)!𝑟! , 𝑛 ≥ 𝑟
𝑛1 !𝑛2 !⋯𝑛 𝑘 !
c. Circular permutation: (𝑛 − 1)!
(𝑛−1)!
d. Special Circular permutation arranged in a key ring or bracelet:
2
21. A boy scout has five signal lights of different colors in a row.
EXERCISES: How many different signals can be made if three lights constitute
a signal?
Evaluate the given expressions.
22. From five engineers and four accountants, find the number of
1. 6! 6. nP0
committees of three that may be formed with two engineers and
2. 100!/97! 7. 6P4 + 6P5 + 6P6
one accountant in each committee?
3. (10 – 4)! 8. 10C9
4. 6P 4 9. nCn
23. How many 7-digit phone numbers can be formed if it cannot
5. 12P9 10. 5C5 + 5C4 + 5C3
begin with 0 or 1?
11. In how many ways can 3 coins fall?
24. How many distinguishable ways can 4 beads be arranged in a
circular bracelet?
12. How many two-digit numbers can be formed from the
integers 1,2,3,4,5, and 6,
a. if repetition of digits is allowed? 25. Daniel must read 3 books from a reading list of 15 books.
b.if repetition of digits is not allowed? How many choices does he have?
c. if the numbers should be greater than 50?
26. How many three-digit odd numbers may be formed from the
13. In how many ways can four girls and three boys be seated in numbers 0, 1,2,3,4,5,6,7,8,9 if repetition of digits is not allowed?
a row of seven chairs with
a. the boys and girls alternating; 27. In how many ways may 6 books in Psychology, 4 books in
b. the boys together and the girls together History, 5 books in Mathematics and 2 books in Sociology be
arranged in a shelf if
14. In how many ways can 10 different books be arranged in a a. they may be placed anywhere
shelf? b. the books in History must stand together
c. the books in Sociology must stand together and the
15. In how many ways can 5 different flags be arranged in a row? Mathematics books must also stand together
16. In how many ways may the letters of the word 28. In how many ways may 3 handkerchiefs, 2 handbags, 1
“ASSESSMENT” be arranged? umbrella and 4 accessory clips be chosen from 7 handkerchiefs, 5
handbags, 4 umbrellas and 8 accessory clips?
17. A girl has 5 t-shirts, 4 maong pants and 3 pairs of rubber
shoes. How many outfits can she form consisting of a t-shirt, a 29. How many ways could five people stand in line if two people
pair of pants and a pair of rubber shoes? refuse to stand next to each other?
18. How many license plates can be manufactured with three 30. In the mega lotto, anyone who picks the correct six numbers
letters followed by three digits? No letter nor digits can be (in any order) wins. With the numbers 1 to 45 available, how
repeated. many combinations are possible? How much should a bettor
invest in order to win the jackpot if a ticket costs P20.
19. In how many ways can the word “STATISTICS” be arranged?