Evaluate The Following Factorials, Permutations, And Combina

Evaluate The Following Factorials Permutations And Combinations

Evaluate the following factorials, permutations, and combinations: (a) 6! (b) (c) (d) 8C4 (e) 10P5 (f) C(3, ). If there are 4 ways to travel from town A to town B, 6 ways from town B to town C, and 5 ways from town C to town D, find the number of different ways to travel from town A to town D. An ice cream parlor offers 31 flavors, 7 toppings, and 5 cone styles; determine the number of possible ice cream cone varieties. The Postal Service is promoting 9-digit zip codes: (a) how many are possible with no restrictions, (b) only even digits, (c) with the first digit not zero and the last digit 9. Determine the number of 6-letter radio station call signs: (a) starting with W, with repetitions allowed; (b) starting with W, ending with Z, repetitions allowed; (c) with no repetitions, starting with W, ending with Z. Find the number of 9-member committees from 27 students; include cases where 3 specific students are on the committee. In a game of musical chairs with 12 children and 11 chairs, find how many ways children can sit. For mail delivery to 16 customers, (a) count arrangements of deliveries; (b) arrangements of 9 deliveries out of 16. Select 6 orchids from 11 for a show; (a) total combinations, (b) arrangements including 3 particular orchids. With a jar of 5 yellow, 7 orange, and 6 red jelly beans, calculate probabilities for selecting 3 beans under various conditions. Draw 5 cards from a 52-card deck; find probabilities of specific suits or combinations. Roll a die 10 times; compute probabilities of specific outcomes. Toss a coin 8 times; find probabilities of all tails, at most 3 tails, or exactly 4 heads. Lastly, with 32 students at a school, find probabilities of different numbers of students skiing, given certain assumptions.

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The comprehensive evaluation of factorials, permutations, and combinations provides foundational insight into counting techniques fundamental to probability and combinatorics. These mathematical tools are integral to solving diverse problems, from simple factorial calculations to complex real-world scenario modeling, including travel routes, product varieties, coding systems, selection processes, and probabilistic assessments.

Factorials in Mathematical Contexts

Factorials, denoted by an exclamation point (!), represent the product of all positive integers up to a given number. For instance, 6! equals 720, computed as 6×5×4×3×2×1. Factorials are pivotal in permutations and combinations formulas, serving as the basis for calculating possible arrangements and selections in various contexts.

Permutations and Combinations

Permutations refer to arrangements where order matters, calculated as P(n, r) = n! / (n - r)! for selecting r objects from n. Combinations denote selections where order is irrelevant, given by C(n, r) = n! / [r! (n - r)!]. These formulas are applied to real problems, such as scheduling, grouping, and counting variants without regard to order.

Travel Routes and Path Counting

Calculating multiple stages in travel involves multiplying the number of ways at each segment. For example, traveling from town A to D via intermediate towns B and C involves multiplying the available routes: 4 from A to B, 6 from B to C, and 5 from C to D, resulting in 4×6×5=120 distinct paths. This illustrates the fundamental principle of counting pathways through compound steps.

Product Varieties and Combinatorial Choices

Determining product varieties, such as ice cream cones, involves multiplying the number of options across categories—flavors, toppings, and cone styles. For an ice cream shop with 31 flavors, 7 toppings, and 5 cone styles, the total varieties equal 31×7×5=1085. This principle underscores the multiplicative counting rule for independent choices.

Zip Code Formation and Restrictions

Zip codes are 9-digit sequences, with total possibilities being 10^9 if digits can repeat. Restrictions, such as using only even digits (0, 2, 4, 6, 8), reduce options per digit to 5, yielding 5^9 possibilities. Further constraints, such as the first digit not zero and the last digit fixed as 9, influence the total counts, calculated by fixing certain digits and allowing others to vary accordingly.

Radio Call Signs and Repetition Constraints

Radio station call signs often consist of 6 letters. When repetitions are allowed, and the first must be W, there are 26 options for each subsequent position, resulting in 1×26^5 possibilities. When the first is W and the last Z, with repetitions, the total becomes 1×26^4×1=26^4. If repetitions are not permitted, permutation formulas apply, considering diminishing choices for each position, with the initial being W and the final Z.

Committee Formation and Subgroup Inclusion

Forming committees uses combinations, such as C(27, 9). Including specific members reduces the problem to choosing remaining members from the rest, e.g., C(24, 6) when 3 particular students are mandated. These calculations exemplify conditional selections in combinatorics.

Permutations in Seating and Scheduling

The problem of children seating in chairs involves permutations considering the number of children and chairs. For 12 children and 11 chairs, arrangements equal the permutations of choosing 11 out of 12, i.e., P(12, 11). Mail delivery scheduling involves permutations or arrangements, with the number of possible sequences corresponding to factorial computations or partial arrangements.

Selection of Orchids and Probabilistic Computations

Choosing orchids entails combinations, e.g., C(11, 6). When certain orchids must be included, the problem reduces to selecting from the rest, e.g., C(8, 3). Jelly bean probability calculations are based on hypergeometric distributions, considering the total bean counts and specific selection criteria, providing insights into probabilities for different event outcomes.

Analysis of Card Hands

Drawing 5 cards from a deck involves hypergeometric probabilities. For example, selecting exactly 2 queens and 2 aces involves calculating the combinations for each group and dividing by total combinations. Probabilities of all hearts, or 3 of a suit, involve similar calculations tailored to specified events.

Dice and Coin Probability Challenges

Rolling a die 10 times requires binomial probabilities, where the likelihood of certain outcomes (e.g., exactly 4 threes) is computed via binomial formulas. Coin toss experiments follow similar principles; calculating probabilities for all tails, at most 3 tails, or exactly 4 heads involves binomial distribution formulas.

School Skiing and Sampling Probabilities

The probability that a certain number of students ski, given a proportion and independence assumption, is modeled using binomial distributions. The probability of exactly 24 students skiing, or all 32, involves applying binomial probability mass functions with p=0.54, the proportion of skiers, and n=32.

Conclusion

Understanding factorials, permutations, and combinations is essential across various fields, including logistics, product design, coding, and probabilistic analysis. Mastery of these concepts enables precise problem-solving and decision-making in contexts requiring quantification of options, arrangements, and likelihoods.

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