# Problems on permutation and combination

• spaghetti3451
In summary, there are a total of 8,190 possible different initials that someone can have if they have at least two, but no more than five, initials. Jor El can choose from 312 different names for his son according to the given rules. A student can choose which questions to answer in the final exam in 3,000 possible ways. If a student can pick any two sections to answer and any three questions within each section, there are a total of 8,000 possible ways to choose which questions to answer. There are 10,737,280 possible computer passwords with a repeated symbol. There are a total of 21,000 odd numbers with distinct digits between 10,000 and 99,999
spaghetti3451

## Homework Statement

a) How many different initials can someone have if a person has at least two, but no more than five, initials? You may assume that each initial is one of the 26 uppercase letters of English, and that letters can be repeated.

b) When attempting to name his son, Jor El has decided on the following rules. i) The name must have exactly five letters ii) the second and fourth letters must be vowels and iii) The third and fifth letters must be the same. Assuming that he can choose from the 26 letters in the English alphabet, how many different names can Jor El pick for his son?

c) A professor designed his final exam as follows: There will be three sections in the exam. Each section has 5 questions. Students have to answer 4 questions from A, 3 from B and only 2 from C. It does not matter in what order the students answer the questions within a section. In how many possible ways can a student choose which questions to answer?

d) A professor designed his final exam as follows: There will be three sections in the exam. Each section has five questions. Students have to pick any two sections to answer, in any order. Within each section, they must choose any three questions. In how many possible ways can a student choose which questions to answer?

e) Computer passwords are to consist of a string of six symbols taken from the digits 0-9 and the lowercase letters. How many computer passwords have a repeated symbol?

f) How many odd numbers between 10000 and 99999 have distinct digits?

g) A classroom has two rows of eight seats each. There are 14 students, 5 of whom always sit in the front row and 4 of whom always sit in the back row. In how many ways can the students be seated?

h) A committee of five people is to be chosen from a club that has 10 male members and 12 female members. How many ways can the committee be formed if it is to contain at least two women?

i) A committee of five people is to be chosen from a club that has ten scientists and eight engineers. How many ways can the committee be formed if it has to contain at least two scientists and at least one engineer?

j) There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the following questions based on this information.
i) How many ways can the committee be formed from the supervisors available?
ii) How many ways can the committee be formed if the General Secretary must be a female?
iii) How many ways can the committee be formed if it must have at least one man and at least one woman?

## The Attempt at a Solution

a) 262 + 263 + 264 + 265
b) 262*52
c) (5C4)(5C3)(5C2)
d) (3C2)(5C3)
e) (36)8 - (36P8)
f) 5*8*7*6*5
g) (8P8)(8P6) + (8P7)(8P7) + (8P6)(8P8)
h) (12C2)(10C3) + (12C3)(10C2) + (12C4)(10C1) + (12C5)(10C0)
i) (10C2)(8C3) + (10C3)(8C2) + (10C4)(8C1)
j) i) 17C3 ii) (7C1)(16C2) iii) (10C1)(7C2) + (10C2)(7C1)

It would be nice if you could check the answers and point out any mistakes.

Check (d) again.
I don't understand your answer at (e) and I think you forgot the zero (or something else) at (f).

I didn't check (g) to (j).

In (e), you seem to have misread the number of symbols as 8.
In (g), you've overlooked that e.g. when there are eight in the front row three of them are variable.
In (j), you have here and there overlooked that there are three distinct roles.

## 1. What is the difference between permutation and combination?

Permutation is the arrangement of objects in a specific order, while combination is the selection of objects without considering the order in which they are chosen.

## 2. How do I know when to use permutation or combination?

Permutation is used when the order of objects is important, such as in arranging a sequence of events. Combination is used when the order does not matter, such as in selecting a group of people for a committee.

## 3. What is the formula for calculating permutations?

The formula for calculating permutations is n! / (n-r)!, where n is the total number of objects and r is the number of objects being arranged.

## 4. What is the formula for calculating combinations?

The formula for calculating combinations is n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects being selected.

## 5. Can permutation and combination be used in real-life situations?

Yes, permutation and combination can be applied in various real-life situations such as arranging seats in a theater, creating unique passwords, and selecting lottery numbers.

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