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In how many ways can a dozen books be placed on four distinguishable shelves

  1. if the books are indistinguishable copies of the same title?
  2. if no two books are the same, and the positions of the books on the shelves matter? [Hint: Break this into $12$ tasks, placing each book separately. Start with the sequence $1, 2, 3, 4$ to represent the shelves. Represent the books by $b_{i}, i = 1, 2,\dots, 12.$ Place $b_{1}$ to the right of one of the terms in $1, 2, 3, 4.$ Then successively place $b_{2}, b_{3},\dots, \text{and }\:b_{12}.]$
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admin asked May 1, 2020
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Use the product rule to prove Theorem $4,$ by first placing objects in the first box, then placing objects in the second box, and so on.