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Homework answers / question archive / Institute of Business Administration - MATH 1021 Chapter 2 Question1)How many 2n-digit positive integers can be formed if the digits in odd positions (counting the rightmost digit as position 1) must be odd and the digits in even positions must be even and positive? Question:2 How many ternary strings of length 2n are there in which the zeroes appear only in odd-numbered positions? 3 Question Mrs

Institute of Business Administration - MATH 1021

Chapter 2

Question1)How many 2n-digit positive integers can be formed if the digits in odd positions (counting the rightmost digit as position 1) must be odd and the digits in even positions must be even and positive?

Question:2 How many ternary strings of length 2n are there in which the zeroes appear only in odd-numbered positions?

3 Question^{ }Mrs. Ste?en’s third grade class has 30 students in it. The students are divided into three groups (numbered 1, 2, and 3), each having 10 students.

a) The students in group 1 earned 10 extra minutes of recess by winning a class competition. Before going out for their extra recess time, they form a single ?le line. In how many ways can they line up?

b) When all 30 students come in from recess together, they again form a single ?le line. However, this time the students are arranged so that the ?rst student is from group 1, the second from group 2, the third from group 3, and from there on, the students continue to alternate by group in this order. In how many ways can they line up to come in from recess?

Question 4 In this exercise, we consider strings made from uppercase letters in the English alphabet and decimal digits. How many strings of length 1010 can be constructed in each of the following scenarios?

- The first and last characters of the string are letters.
- The first character is a vowel, the second character is a consonant, and the last character is a digit.
- Vowels (not necessarily distinct) appear in the third, sixth, and eighth positions and no other positions.
- Vowels (not necessarily distinct) appear in exactly two positions.
- Precisely four characters in the string are digits and no digit appears more than one time.

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Question 5 Let X be the set of the 26 lowercase English letters and 10 decimal digits. How many X-strings of length 15 satisfy *all* of the following properties (at the same time)?

- The first and last symbols of the string are distinct digits (which may appear elsewhere in the string).
- Precisely four of the symbols in the string are the letter 't'.
- Precisely three characters in the string are elements of the set V={a,e,i,o,u} and these characters are all distinct

Question 6

An ice cream shop has a special on banana splits, and Xing is taking advantage of it. He's astounded at all the options he has in constructing his banana split:

- He must choose three different flavors of ice cream to place in the asymmetric bowl the banana split is served in. The shop has 20 flavors of ice cream available.
- Each scoop of ice cream must be topped by a sauce, chosen from six different options. Xing is free to put the same type of sauce on more than one scoop of ice cream.
- There are 1010 sprinkled toppings available, and he must choose three of them to have sprinkled over the entire banana split.

- How many different ways are there for Xing to construct a banana split at this ice cream shop?
- Suppose that instead of requiring that Xing choose exactly three sprinkled toppings, he is allowed to choose between zero and three sprinkled toppings. In this scenario, how many different ways are there for him to construct a banana split?

Question 7 How many integer-valued solutions are there to each of the following equations and inequalities?

- x1+x2+x3+x4+x5=63,x1+x2+x3+x4+x5=63, all xi>0xi>0
- x1+x2+x3+x4+x5=63,x1+x2+x3+x4+x5=63, all xi≥0xi≥0
- x1+x2+x3+x4+x5≤63,x1+x2+x3+x4+x5≤63, all xi≥0xi≥0
- x1+x2+x3+x4+x5=63,x1+x2+x3+x4+x5=63, all xi≥0,xi≥0, x2≥10x2≥10
- x1+x2+x3+x4+x5=63,x1+x2+x3+x4+x5=63, all xi≥0,xi≥0, x2≤9

^{ }

**Questio 8 **14 How many integer solutions are there to the inequalityx1+x2+x3+x4+x5≤782x1+x2+x3+x4+x5≤782provided that x1,x2>0,x1,x2>0, x3≥0,x3≥0, and x4,x5≥10?

**Question** 9 Give a combinatorial argument to prove the identity

k(nk)=n(n−1k−1).

**Question** 10 How many lattice paths are there from ((0,0) to (10,12)?

**Question**: 11 how many lattice paths from (0,0) to (14,73) are there that do not pass through (6,37)?

**Question: 12 **The setting for this problem is the fictional town of Mascotville, which is laid out as a grid. Mascots are allowed to travel only on the streets, and not “as the yellow jacket flies.” Buzz, the Georgia Tech mascot, wants to go visit his friend Thundar, the North Dakota State University mascot, who lives 66 blocks east and 77 blocks north of Buzz's hive. However, Uga VIII has recently moved into the doghouse 22 blocks east and 33 blocks north of Buzz's hive and already has a restraining order against Buzz. There's also a pair of tigers (mother and cub) from Clemson who live 11 block east and 22 blocks north of Uga VIII, and they're known for setting traps for Buzz. Buzz wants to travel from his hive to Thundar's pen every day without encountering Uga VIII or The Tiger and The Tiger Cub. However, he wants to avoid the boredom caused by using a route he's used in the past. What is the largest number of consecutive days on which Buzz can make the trip to visit Thundar without reusing a route (you may assume the routes taken by Buzz only go east and north)?

**Question: 13 ** Determine the coefficient on x12y24x12y24 in (x3+2xy2+y+3)18.(x3+2xy2+y+3)18. (Be careful, as xxand yy now appear in multiple terms!)

Question 14 How many ways are there to paint a set of 2727 elements such that 77 are painted white, 66 are painted old gold, 22 are painted blue, 77 are painted yellow, 55 are painted green, and 00 of are painted red?

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