Finite Math Quiz 2 Enumeration Read The F

Finite Math Quiz 2 Enumeration Read The F

Finite Math Quiz 2 Enumeration Read The F

Read the following scenarios and determine which method would be the best approach to solve the problem. Each problem involves selecting or arranging items, counting possibilities, or determining the number of outcomes using combinatorial methods.

Paper For Above instruction

1. A coach needs to select 15 seniors to play on the football team. How many different ways can the coach select the team if 37 seniors try out?

Solution: This scenario involves choosing a subset of seniors without regard to order; thus, it is a combination problem. The number of ways is calculated using the combination formula C(n, r) = n! / [r! (n - r)!], where n=37 and r=15.

Number of ways = C(37, 15) = 37! / (15! * 22!).

2. A fighting game has 12 different fighters to choose from. The player then fights the remaining 11 characters in random order. How many sequences of fights are possible?

Solution: Since the sequence of fighters matters and no fighter is repeated, this is a permutation problem. The total sequences possible are permutations of 12 fighters taken 12 at a time, i.e., P(12, 12)= 12!.

3. A sales representative has to visit 7 different clients and can choose to visit them in any order. How many different routes can the sales representative take?

Solution: The order matters, and each client is visited once, so this is a permutation problem. Total routes = 7!.

4. A fast food restaurant has 12 combo meals. Each combo meal has 7 options for a side dish, 5 types of drink, and is available in 2 sizes (“Regular” or “Large”). How many different combo meals can be purchased?

Solution: Use the rule of product: multiply the options for each component. Total combos = 12 7 5 * 2.

5. A used video store has 35 kung-fu movies, 73 dramatic movies, 27 musicals, and 43 sci-fi movies. How many total videos are available for purchase?

Solution: Sum all categories: total videos = 35 + 73 + 27 + 43.

6. How many different values can be represented by two bytes? Note a byte consists of 8 bits, with each bit capable of storing 0 or 1.

Solution: Each bit has 2 options; total combinations are 2⁸ per byte, so for two bytes: 2⁸ * 2⁸ = 2¹⁶ possible values.

7. Compute the following permutations and combinations:

  • $ P 15,
  • $ C 10,
  • $ P 13,
  • $ C 18,17

Solutions:

  • P(15) = P(15, 15) = 15!
  • C(10) = C(10, 10) = 1
  • P(13, 13) = 13!
  • C(18, 17) = 18! / (17! * 1!) = 18

11. How many different 5-card poker hands containing only diamonds are possible?

Solution: Select 5 diamonds from 13 total diamonds: C(13, 5).

12. The North American Numbering Plan (NANP) has established a protocol for how area codes are numbered. The first digit can be 2–9, and the second and third digits can be 0–9. How many area codes are possible?

Solution: Choices: 8 options for first digit (2–9), 10 options each for second and third digit: 8 10 10 = 800.

13. A state uses two letters followed by three digits for license plates. How many plates can be created?

Solution: 26 options per letter (assuming English alphabet), so total = 26 26 10 10 10.

14. A tower defense game requires placing 10 towers in order, with 3 red, 5 grey, and 2 green towers to choose from. How many arrangements are possible?

Solution: Permutations of multiset: total arrangements = 10! / (3! 5! 2!).

15. A campus coffee shop offers 15 hot beverages, 8 sandwiches, 12 cold beverages, and 11 bagels. How many meals are possible if choosing either hot beverage and a bagel, or cold beverage and a sandwich?

Solution: Number of combinations: (hot beverage bagel) + (cold beverage sandwich) = (15 11) + (12 8).

16. A puzzle requires creating a three-color code using only Red, Blue, and Yellow, with colors allowed in any position and repetition allowed. How many combinations?

Solution: 3 options per position, total = 3³ = 27.

17. A travel app suggests 3 hotels and 5 restaurants at random in a town with 8 hotels and 18 restaurants. How many different suggestions are possible?

Solution: Number of suggestions = 3 5, but since choices are from the total available, total suggestions = 8 7 18 17 (assuming choosing different hotels and restaurants). Since the problem states 3 hotels and 5 restaurants, likely the total combinations are 8 7 for hotels and 18 17 for restaurants, but it depends on the interpretation. Assuming no restriction, total combinations = 8 7 18 * 17.

18. The World Series matchups possible between teams from the American League (14 teams) and National League (16 teams): how many matchups?

Solution: Each American League team can face each National League team, so total = 14 * 16.

19. Characters with a Tria Nomina style: Praenomina (17 options), Nomen (130 options), Cognomen (193 options). How many unique names?

Solution: Total = 17 130 193.

20. How many ways to arrange 15 billiard balls in a triangular rack?

Solution: Arranged in order, total arrangements = 15!.

References

  • Brualdi, R. A. (2010). Introductory Combinatorics. Pearson.
  • Grinstead, C., & Snell, J. (1997). Introduction to Probability. American Mathematical Society.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Ross, S. (2014). A First Course in Probability. Pearson.
  • Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley.
  • Stein, S. K., & Szabo, M. (2011). Discrete Mathematics for Computer Science. Chapman & Hall/CRC.
  • Honsberger, R. (1991). Mathematical Gems II. Mathematical Association of America.
  • Mitzenmacher, M., & Upfal, E. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press.
  • Vanderbilt, J. (2011). Discrete Mathematics. Cengage Learning.
  • Knuth, D. E. (1998). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.

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