Homework Week 11

Problem 1

List all permutations of \(\{A,B,C\}\).

Problem 3

How many permutations of \(\{A,B,C,D,E,F,G\}\) end with \(A\)?

Problem 6

Find the value of each:

1. \(\binom51\)
   
2. \(\binom53\)
   
3. \(\binom84\)
   
4. \(\binom88\)
   
5. \(\binom80\)
   
6. \(\binom{12}{6}\)

Problem 7

Find the number of 5-permutations of a set with 9 elements.

Problem 13

A group contains \(n\) men and \(n\) women. How many ways are there to arrange these people in a row if the men and women alternate?

Problem 19

A coin is flipped 10 times. How many possible outcomes:

1. are there in total?
   
2. contain exactly two heads?
   
3. contain at most three tails?
   
4. contain the same number of heads and tails?

Problem 23

How many ways are there for 8 men and 5 women to stand in a line so that no two women stand next to each other?

Problem 25

How many ways are there for 4 men and 5 women to stand in a line so that:

1. all men stand together?
   
2. all women stand together?

Bit Strings

How many bit strings of length 10 contain exactly four 1s?

How many bit strings of length 10 contain at most four 1s?

How many bit strings of length 10 contain at least four 1s?

How many bit strings of length 10 contain an equal number of 0s and 1s?

Blocks and Permutations

How many permutations of \(ABCDEFG\) contain the string \(BCD\)?

How many permutations of \(ABCDEFG\) contain \(B\), \(C\), and \(D\) together in any order?

How many permutations of \(ABCDEFG\) contain the string \(CFGA\)?

How many permutations of \(ABCDEFG\) contain the strings \(BA\) and \(GF\)?

How many permutations of \(ABCDEFG\) contain \(BA\) or \(GF\)?

Gaps, Roles, Committees, Strings

How many ways are there to line up 7 men and 3 women so that no two women are next to each other?

A club has 25 members. How many ways are there to choose a president, vice president, secretary, and treasurer, if no person can hold more than one office?

A committee of 4 people is chosen from 25 members. How many ways are there?

How many strings of length 6 can be formed from uppercase English letters if letters can repeat?

How many strings of length 6 can be formed from uppercase English letters if letters cannot repeat?

How many strings of length 6 can be formed from uppercase English letters if letters cannot repeat and the first letter must be A?

Problem 1

Find the expansion of \((x+y)^4\):

1. using combinatorial reasoning
   
2. using the binomial theorem

Problem 37

Count the number of paths from \((0,0)\) to \((m,n)\), where each step is either one unit right or one unit up.

Problems 1, 3, 7, 9

Exact text not pasted in chat. Listed in the official Week 11 assignment for Section 6.6.

Problem 5

Find the next larger permutation in lexicographic order after:

1. 1432
   
2. 54123
   
3. 12453
   
4. 45231
   
5. 6714235
   
6. 31528764

Combination Lexicographic Practice

Practice finding the next combination given any \(r\)-combination of \(\{1,2,3,\dots,n\}\).