Probability

1) A drawer contains n=5 different and distinguishable pairs of socks (a total of 10 socks). If a person randomly selects 4 socks, what is the probability that there is no matching pair in the sample? 2) A drawer contains n different and distinguishable pairs of socks (a total of 2n socks). A person randomly selects 2r of the socks, where 2r > n. In terms of n and r, what is the probability that there is no matching pair in the sample? I'm totally clueless on how to go about doing this, if you could explain each step, I'd greatly appreciate it!

Linear programming-maximze advertising budget

As part of a campaign to promote its annual clearance sale, the Excelsior Company decided to buy television advertising time on Station KAOS. Excelsior's advertising budget is $102,000. Morning time costs $3000/minute, afternoon time costs $1000/minute, and everning (prime) time costs $12,000 minute. Becasue of pervious commitments, KAOS cannot offer Excelsior more than 6 min of prime time or more than a total of 25 min of advertising time over the 2 weeks in which the commercials are to be run. KAOS estimates that morning commericials are seen by 200,000 people, afternoon commericals are seen by 100,000 people and everning commericals are seen by 600,000 people. How much morning, afternoon,... click for more

Formulate, then solve the dual problem by the simplex method and analyze the duality

A farmer grows two crops, corn and soybeans. Suppose that each acre planted with corn requires 1 man day of labor, $3 investment and yields a profit of $35 at harvest. Each acre of soybeans on the other hand requires 2 man days of labor,$1 investment and returns a profit of $20 at harvest. The farmer had a total of 70 acres available for these crops. He has $140 he can invest and 80 man days of labor available. How many arces of each crop should he plant for maximum profit?

Formulate then solve the dual problem by the simplex method and analyze the duality

A nutritionalist is preparing a supplement mixture to add to a preparation of oatmeal. The nutritionalist has 3 mixtures available that can be combined to give the desired mixture. Each gram of the first mixture contains 1 unit of calcium, 1 unit of iron, and cost 70 cents. Each gram of the second mixture contains 1 unit of calcium, 2 units of iron, and costs 80 cents. Each gram of the third mixture contains 3 units of calcium, 1 unit of iron, and cost $1.40. Each batch of the product needs 35 units of calcium and 20 units of iron. How much of each mixture should be added to each batch of the product to meet these requirements at minimum cost?

It is dealing with finite automatons

(a)For each of the following languages over the unary alphabet {a}, construct a finite automaton accepting it. i. {a^2} ii. {a^2, a^3, a^4} (b) Let A be any finite nonempty subset of {a, a^2, a^3, a^4,...}. Is there always a finite automaton that accepts A?