dis7-sol.pdf - CS 70 Fall 2019 1 Discrete Mathematics and Probability Theory Alistair Sinclair and Yun S Song DIS 7 Farmer’s Market Suppose you want k

# dis7-sol.pdf - CS 70 Fall 2019 1 Discrete Mathematics and...

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CS 70 Discrete Mathematics and Probability Theory Fall 2019 Alistair Sinclair and Yun S. Song DIS 7 1 Farmer’s Market Suppose you want k items from the farmer’s market. Count how many ways you can do this, assuming: (a) There are pumpkins and apples at the market. (b) There are pumpkins, apples, oranges, and pears at the market (c) There are n kinds of fruits at the market, and you want to end up with at least two different types of fruit. Solution: This is a classic “balls and bins” (also known as “stars and bars”) problem. (a) k + 1 . (b) ( k + 3 3 ) . (c) There are ( n + k - 1 n - 1 ) ways to choose k fruits of n types with no additional restrictions. n of these combinations, however, contain only one variety of fruit, so we subtract them for a total of ( n + k - 1 n - 1 ) - n . 2 Counting Practice (a) If you shuffle two (identical) decks of cards together, you get a stack of 104 cards, where each different card type is included twice. How many different ways are there to order this stack of cards? (b) How many different anagrams of GHOST are there if: (1) H is the right neighbor of G; (2) G is on the left of H (and not necessarily H’s neighbor)? (c) There are 20 socks in a drawer, none of which match. How many different ways are there to pair up these socks? (Assume that any sock can be paired with any other sock.) Solution: CS 70, Fall 2019, DIS 7 1
(a) If we consider the 104! rearrangements of 2 identical decks, since each card appears twice, we would have overcounted each distinct rearrangement. Consider any distinct rearrangement of the 2 identical decks of 52 cards and see how many times this appears among the rearrangement of 104 cards where each card is treated as different. For each identical pair (such as the two

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