3. Plonk Winery are packing \"mixed dozens" for an upcoming sale. They have available 3 red varieties (Shiraz, Merlot and Cabernet) and 2 white varieties (Riesling and Verdelho). Each mixed dozen must contain 12 bottles, at least two of which must be red, and at least two must be white. (a) How many mixed dozens are possible with these constraints? (b) How many mixed dozens include at least one bottle of each of the 5 varieties? (c) Towards the end of the day, the supply of Merlot runs low. How many mixed dozens are there that contain at most one bottle of Merlot? 4. The Scarlet Buttery has a lifespan of at most 2 weeks. Suppose that during those two weeks its probability of dying has a probability density function (PDF) given by 32 2 z(:::2 + yg) where w is a continuous variable measuring the lifespan in weeks, and 1:, y, z E R. (a Express 2 as a function of 1: and y. (c Check that your answer to (b) produces a valid PDF. (d For the PDF that you found in (b), what is the probability that a Scarlet Buttery lives for at least 1 week? (e) What is the expected lifespan of a Scarlet buttery? ) (b) Which values of z and y make 2 as large as it can be? ) l