# Day 9 more problems .pdf

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Combinatorial Analysis

Group homework: Look for a partner and solve the following ten

problems. They are due Tuesday, November 5.

Problem 1 A bakery boasts eight varieties of doughnuts. If a box of doughnuts contains one dozen, how many different options are there for a box of

doughnuts?

Problem 2 What is the number of nondecreasing sequences of length r whose

terms are taken from 1,2,...,k?

Problem 3 Continuing with the doughnut example, what are the number of

different options for boxes of doughnuts containing at least one doughtnut of

each of the eight varieties?

Problem 4 What is the number of integral solutions of the equation x1 + x2 +

· · · + x4 = 20, in which x1 ≥ 3, x2 ≥ 1, x3 ≥ 0, x4 ≥ 5?

Problem 5 Find the number of nonnegative integer solutions for the inequality

x1 + x2 + x3 + x4 < 19.

Problem 6 In how many ways can 12 indistinguishable apples and 1 orange

be distributed among three children in such a way that each child gets at least

one piece of fruit?

Problem 7 Determine the number of ways to distribute 10 orange drinks, 1

lemon drink, and 1 lime drink to four thirsty students so that each student gets

at least one drink, and the lemon and lime drinks go to different students.

Problem 8 A pizzeria has 10 different types of toppings. If you need to order

13 one-topping pizzas for a party, how many different orders could you place?

How about if I want at least one pizza of each topping?

Problem 9 There are 2n + 1 identical books to be put in a bookcase with three

shelves. In how many ways can this be done if each pair of shelves together

contain more books that the other shelf ?

Problem 10 What is the number of integral solutions of the equation

x1 + x2 + x3 + x4 = 18

that satisfy 1 ≤ x1 ≤ 5, −2 ≤ x2 ≤ 4, 0 ≤ x3 ≤ 5, 3 ≤ x4 ≤ 9?

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