For each of the following questions, write your solutions in terms of binomial coefficients. Do not...

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For each of the following questions, write your solutions in terms of binomial coefficients. Do not evaluate your answers. For credit, you must show your work and explain your answer. We throw m balls into n bins; each bin can fit multiple balls. We distinguish one configuration from another only by the content of the bins, not by the order in which the balls land in the bins. (a) (1 point) Suppose the m balls and the n bins are distinguishable, i.e., each ball is labeled with a different number and each bin is given a distinct name. How many ways are there to throw m balls into n bins? (b) (1 point) Suppose the m balls and n bins are distinguishable, that n > m, and that we would like each bin to have at most 1 ball. How many ways are there to throw the balls? (c) (1 point) Suppose the m balls and n bins are distinguishable; how many ways can we throw the balls if we want the first and the second bins to have exactly two balls? (d) (2 points) Same premise as the above problem, but now m = 2n and we would like every bin to have exactly 2 balls. How many ways are there to throw the balls? (e) (2 points) Generalize the previous problem. Suppose r +12+ +rn = m, where r are positive integers. We would like the ith bin to have ri balls. How many ways are there to throw the balls? (f) (2 points) Now suppose that the balls are identical and hence indistinguishable, i.e., having ball A in bin 1 and ball B in bin 3 is now considered the same as having ball A in bin 3 and ball B in bin 1. How many ways are there to throw the balls such that the first bin has at most 2 balls and all other bins have at least 3? Assume there are enough balls. For each of the following questions, write your solutions in terms of binomial coefficients. Do not evaluate your answers. For credit, you must show your work and explain your answer. We throw m balls into n bins; each bin can fit multiple balls. We distinguish one configuration from another only by the content of the bins, not by the order in which the balls land in the bins. (a) (1 point) Suppose the m balls and the n bins are distinguishable, i.e., each ball is labeled with a different number and each bin is given a distinct name. How many ways are there to throw m balls into n bins? (b) (1 point) Suppose the m balls and n bins are distinguishable, that n > m, and that we would like each bin to have at most 1 ball. How many ways are there to throw the balls? (c) (1 point) Suppose the m balls and n bins are distinguishable; how many ways can we throw the balls if we want the first and the second bins to have exactly two balls? (d) (2 points) Same premise as the above problem, but now m = 2n and we would like every bin to have exactly 2 balls. How many ways are there to throw the balls? (e) (2 points) Generalize the previous problem. Suppose r +12+ +rn = m, where r are positive integers. We would like the ith bin to have ri balls. How many ways are there to throw the balls? (f) (2 points) Now suppose that the balls are identical and hence indistinguishable, i.e., having ball A in bin 1 and ball B in bin 3 is now considered the same as having ball A in bin 3 and ball B in bin 1. How many ways are there to throw the balls such that the first bin has at most 2 balls and all other bins have at least 3? Assume there are enough balls.