Evaluate each number. $$ C(4,3) $$

Minimalist Art You are exhibiting your collection of minimalist paintings. Art critics have raved about your paintings, each of which consists of 10 vertical colored lines set against a white background. You have used the following rule to produce your paintings: Every second line, starting with the first, is to be either blue or grey, while the remaining five lines are to be either all light blue, all red, or all purple. Your collection is complete: Every possible combination that satisfies the rules occurs. How many paintings are you exhibiting?

You are packing for a short trip and want to take 2 of the 10 shirts you have hanging in your closet. Critique the following decision algorithm and calculation of how many different ways you can choose two shirts to pack: Step 1 , choose one shirt, 10 choices. Step 2, choose another shirt, 9 choices. Hence there are 90 possible choices of two shirts.

Suppose you are a salesperson who must visit the following 23 cities: Dallas, Tampa, Orlando, Fairbanks, Seattle, Detroit, Chicago, Houston, Arlington, Grand Rapids, Urbana, San Diego, Aspen, Little Rock, Tuscaloosa, Honolulu, New York, Ithaca, Charlottesville, Lynchville, Raleigh, Anchorage, and Los Angeles. Leave all your answers in factorial form. a. How many possible itineraries are there that visit each city exactly once? b. Repeat part (a) in the event that the first five stops have already been determined. c. Repeat part (a) in the event that your itinerary must include the sequence Anchorage, Fairbanks, Seattle, Chicago, and Detroit, in that order.

Which of the following represent permutations? (A) An arrangement of books on a shelf (B) A group of 10 people in a bus (C) A committee of 5 senators chosen from 100 (D) A presidential cabinet of 5 portfolios chosen from 20

If \(A\) and \(B\) are finite sets with \(A \subset B\), how are \(n(A)\) and \(n(B)\) related?