A knight and knave problem

Three inhabitants A, B, C are being interviewed. A and B make the following statements:

A: B is a knight

B: If A is a knight, so is C

The algebraic solution is very simple but the informal explanation is much more interesting

A = B

B = A?C

B = ?A ú C

A = ?A ú C

?A =?(?A ú C)

?A = A ù ?C

?A ù ?A = ?A ù (A ù ?c)

?A = F,

A = T or A is a knight

B = A = T or B is a knight

Since A = T and A->C, C = T and C is a knight

That is, A, B, and C are knights

The informal solution

A reasonable solution

Suppose A is knight. Then A’s statement is true, namely, B is a knight. Since B is a knight, B’s statement is true and, since A is knight, B’s statement’s antecedent is also true. Therefore, by the law of detachment, C is a knight. In other words, if A is knight then C is a knight. But this is B’s statement and it is true, therefore B is a knight. But A claims that B is a knight and therefore A’s claim is true and A is knight. That is A, B, and C are knights

Pascal’s Triangle

Binomials

Multinomials

Suppose we have a box containing five upper case letters and three lower case letters. How many ways can we select one upper case letter and one lower case letter? How many ways can we select two upper case letters and two lower case letters?

Suppose there is one upper case and one lower case letter. How many ways can you select and upper case and a lower case letter?

Suppose there are four paths to the top of the mountain and back down again. How many ways are there up the mountain and back down again?

Suppose you had to go down a different way than the way up?

If a penny and a nickel be tossed, in how many ways can they fall (either heads or tails)?

If two dice be thrown together in how many ways can they fall (one, two, three, four, five and six)?

In how many ways can two prizes be given to a class of ten boys, without giving both to the same boy?

In how many ways can two prizes be given to a class of ten boys, it being permitted to give both prizes to the same boy?

Two persons get into a railway carriage where there are six vacant sets. In how many different ways can they seat themselves?

In how many ways can we make a two-letter word out of an alphabet of twenty-six letters, the two letters in the word being different?

Suppose you remove the “different letter” restriction. How many ways?

In how many ways can we select a consonant and a vowel out of an alphabet of twenty consonants and six vowels?

How many 2 letters “words” that have exactly one consonant and one vowel?

How many two letter “words” can you form from the set {a,b}?

ab and ba

In how many ways can we make a two-letter word, consisting of one consonant and one vowel?

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(4)4 = 4 x 3 x 2 x 1 = 24

A table being laid for six persons, in how many ways can they take their places?

In how many ways can six children form themselves into a ring?

How many three-letter words could be made out of an alphabet of twenty-six letters, not using any letter more than once?

How many words, not using any letter more than once can you generate from this set?

In how many ways can three boys divide twelve oranges, each taking four?

Out of one hundred things, in how many ways can three things be selected?

Out of a basket of twenty pears at three a penny, how many ways are there of selecting six pennies worth of pears?

In how many ways can the same choice be exercised as to include the largest pear?

In how many ways can the same choice be exercised without taking the smallest pear?

Out of forty-two liberals and fifty conservatives, what choice is there in selecting a committee consisting of four liberals and four conservatives?

A company of volunteers consists of a captain, a lieutenant, an ensign, and eight rank and file. In how many ways can ten men be selected so as to include the captain?

In how many ways can ten men be selected so as to include at least one officer?

In how many ways can ten men be selected to include exactly one officer?

From a set of five pairs of shoes, two of the shoes are selected at random. Find the probability of each of the following:

both are from the same pair

one left shoe and one right shoe are selected

Out of twenty men and six women, what choice have we in selected three men and three women?

Out of twenty consonants and six vowels, in how many ways can we make a word, consisting of three different consonants, and three different vowels?

Out of twenty-six letters of the alphabet, in how many ways can we make a word consisting of four different letters, one of which must be always a?

Out of the twenty-six letters of the alphabet, in how many ways can we make a word consisting of four different letters, two of which must be a and b?

Out of twenty consonants and six vowels, in how many ways can we make a word consisting of three different vowels and two different consonants, one of the vowels being always a?

In how many ways can an arrangement of four letters be made out of the letters of the words choice and chance? The different ways of selecting the four letters, may be classified as follows: all four alike, three alike and one different, two alike and two others alike, two alike and the other two different, all four different

In how many ways can an arrangement of three things be made out of fifteen things, of which five are of one sort, four of another sort, three of another sort, and the remaining three of another sort?

A room has six doors. In how many ways is it possible to enter by one d or and leave by another?

A tire store carries eight different sizes of tires, each in both tube and tubeless variety, each with either nylon or rayon cord, and each with white sidewalls or plain black. How many different kinds of tires does the store have?

How many integers between 100 and 999 have distinct digits?

How many even numbers between 100 and 999 have distinct digits?

In how many ways can ten persons be seated in a row so that a certain two of them are not next to each other?

A college has 720 students. In how many ways can a delegation of ten be chose to represent the college?

A man works in a building located seven blocks east and eight blocks north of his home. Thus in walking to work each day he goes fifteen blocks. All the streets in the rectangular pattern are available to him for walking. In how many different ways can he go from home to work?

How many integers greater than 5300 have the following two properties: the digits of the integer are distinct and the digits 0 and 9 do not occur in the number?

How many distinct permutations of the word “feed” are there?

How many ways can three cards be selected at random from a standard deck of 52 cards so that all the cards are of the same suit?

How many ways can three cards be selected at random from a standard deck of 52 cards so that all the cards are of different suits?

How many ways can five cards be selected at random from a standard deck of 52 cards so that there exactly 1 set of 2 of a kind?

How many ways can five cards be selected at random from a standard deck of 52 cards so that there are exactly 2 sets of 2 of a kind?

How many ways can you flip 7 coins so that 4 land on heads and 3 land on tails?

A biologist is studying patterns of male (M) and female (F) children in families. A family type is designated by a code; for example, FMM denotes a family of three children of which the oldest is female and the other two males. Note that FMM, MFM, and MMF are different types. How many family types are there among families with at least one but not more than 7 children?

How many permutations of “Mississippi” are there?

How many ways can you flip 7 coins so that 4 land on one side and 3 land on the other side?

How many different sums of money can be made up using one or more coins selected from a cent, a nickel, a dime, a quarter, a half dollar, and a silver dollar?

A man works in a building located seven blocks east and eight blocks north of his home. Thus in waling to work each day he goes fifteen blocks. All the streets in the rectangular pattern are available to him for walking. In how many different ways can he go from home to work walking only fifteen blocks?

How many different permutations are there of the letters of the word Mississippi taken all at a time? In other words, in how may different orders is it possible to write the letters of the word Mississippi?

An examination consists of ten questions, of which a student is to answer eight and omit two. In how many ways can a student make his selection?

The mathematics departments consists of 8 men and 4 women. How many committees of 6 may be chosen without restriction? How many committees of 6 may be chosen that contains 3 men and 3 women? How many committees of 6 may be chosen that contains at least one man?