4.6 Permutations

Explanation
The question asks to evaluate 6!. The exclamation mark denotes the factorial function, which means multiplying a number by all the positive integers less than it. Therefore, 6! is calculated as 6 x 5 x 4 x 3 x 2 x 1, which equals 720.

Explanation
The expression P(7, 3) represents the permutation of selecting 3 items from a set of 7 items in a specific order. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the number of items in the set and r is the number of items being selected. In this case, P(7, 3) = 7! / (7-3)! = 7! / 4! = (7 * 6 * 5 * 4!) / 4! = 7 * 6 * 5 = 210. Therefore, the correct answer is 210.

Explanation
There are 4 people to be arranged in a straight line. The first person can be chosen in 4 ways, the second person can be chosen in 3 ways (as one person is already placed), the third person can be chosen in 2 ways (as two people are already placed), and the fourth person can be chosen in 1 way (as three people are already placed). Therefore, the total number of ways to arrange the four people is 4 x 3 x 2 x 1 = 24.

Explanation
The word "ZAMBONI" has 7 letters. To find the number of different arrangements, we use the formula for permutations of a set, which is n!, where n is the number of elements in the set. In this case, n = 7. Therefore, the number of different arrangements of the letters in the word "ZAMBONI" is 7! = 5040.

Explanation
The word "ROLLER" has 6 letters. To find the number of different arrangements, we can use the formula for permutations. Since all the letters are unique, we can arrange them in 6! (6 factorial) ways, which is equal to 720. However, since the letter "L" appears twice, we need to divide the total number of arrangements by 2! (2 factorial) to account for the repeated arrangement. Therefore, the number of different arrangements of the letters in the word "ROLLER" is 720 / 2 = 360.

Explanation
There are 5 people lined up for the race, and we need to determine the number of different ways the first and second positions can be assigned. Since the order matters, we can use the permutation formula. The number of ways to choose the first position is 5, and once the first position is assigned, there are 4 people remaining for the second position. Therefore, the total number of different ways is 5 * 4 = 20.

Explanation
There are four terminals on the thermostat, but only three wires coming from the wall. This means that one of the terminals will not be connected to any wire. The wire can be connected to any of the four terminals, so there are four possible options for the first wire. After the first wire is connected, there are three remaining terminals for the second wire to be connected to. Finally, there are two remaining terminals for the third wire to be connected to. Therefore, the total number of different ways the wires can be connected is 4 x 3 x 2 = 24.

Explanation
There are 6 letters in the word FERMAT. Out of these, 3 letters (E, A, and T) are vowels. In order for an arrangement to end in a vowel, the last letter must be one of these 3 vowels. Therefore, there are 3 options for the last letter. The remaining 5 letters can be arranged in any order. Using the formula for permutations, we can calculate the total number of arrangements as 3 options for the last letter multiplied by the number of permutations of the remaining 5 letters (5!). This gives us a total of 3 * 5! = 3 * 120 = 360. However, since the question specifies that each arrangement must end in a vowel, we need to subtract the number of arrangements where the last letter is not a vowel. There are 3 consonants in the word FERMAT, and the remaining 4 letters can be arranged in any order. Therefore, there are 3 * 4! = 3 * 24 = 72 arrangements where the last letter is not a vowel. Subtracting this from the total number of arrangements, we get 360 - 72 = 288. However, this calculation includes arrangements where the first letter is a vowel, which is not allowed according to the question. There are 2 vowels (E and A) that can be in the first position, and the remaining 4 letters can be arranged in any order. Therefore, there are 2 * 4! = 2 * 24 = 48 arrangements where the first letter is a vowel. Subtracting this from the total number of arrangements, we get 288 - 48 = 240. So, there are 240 arrangements of the letters in FERMAT if each arrangement must end in a vowel.

Explanation
Consider QWERTY as a single item. 5!/[(2!)(2!)]

Explanation
9! x 2! (consider the two coaches as a single item and then consider the different arrangements of the two coaches after)