Quiz: Discrete Math (Counting)

Explanation
There are 900 4-digit counting numbers that are multiples of 10. To find this, we need to determine the range of 4-digit numbers (1000 to 9999) and then calculate how many of these numbers are divisible by 10. Since every number that ends with a 0 is divisible by 10, we can count the number of possibilities by finding the number of different choices for the thousands, hundreds, and tens place (which can be any digit from 0 to 9) and fix the units place as 0. Therefore, there are 10 choices for each place (except the units place) resulting in a total of 10*10*10 = 1000 possibilities. However, we need to subtract the case where the thousands place is 0, which gives us 10*10*10 - 10 = 900 4-digit counting numbers that are multiples of 10.

Explanation
When a coin is tossed four times, each toss has two possible outcomes: either heads or tails. Since there are four tosses, the total number of possible outcomes can be calculated by multiplying the number of outcomes for each toss together. In this case, it is 2 x 2 x 2 x 2 = 16. Therefore, there are 16 possible outcomes when a coin is tossed four times.

Explanation
There are four shirts and four pairs of trousers, and each shirt can be worn with any pair of trousers. Therefore, for each shirt, there are four possible combinations with the trousers. Since there are four shirts, the total number of combinations is 4 multiplied by 4, which equals 16.

Explanation
The student has four options for transportation to and from school: Pafra jeep, Bachelor bus, van, or single habalhabal. However, if he prefers to only ride via van to go home, then he can only choose the van as his mode of transportation. Therefore, there is only one possible way for him to travel to and from school, which is by riding the van. Hence, the correct answer is 1.

Explanation
There are 4 ways to choose a man for the first vacancy and 12 ways to choose a woman for the second vacancy. Since these choices are independent, the total number of ways to fill the vacancies is 4 * 12 = 48.

Explanation
There are 4 ways to fill the first vacancy with a man. For the second vacancy, there are 4 men and 12 women to choose from, so there are 16 possible combinations. However, since the second vacancy can be filled by either a man or a woman, we need to subtract the case where both vacancies are filled by men, which is 4. Therefore, the total number of ways to fill the vacancies is 16 - 4 = 12. Since each combination has 2 different orders (man-woman or woman-man), we multiply 12 by 2 to get 24. Finally, since there are 3 different pairs of vacancies that can be filled, we multiply 24 by 3 to get 72.

Explanation
To form a three-digit number greater than 600 using the given digits without repeating any digit, we need to consider the hundreds place first. Since the number needs to be greater than 600, the hundreds place can only be 6 or 8. Once we choose the hundreds place, we have 4 options for the tens place (1, 2, 5, or 8) and 3 options for the units place (the remaining digits). Therefore, the total number of three-digit numbers greater than 600 that can be formed is 2 (choices for the hundreds place) multiplied by 4 (choices for the tens place) multiplied by 3 (choices for the units place), which equals 24.

Explanation
To form a four-digit number, we have four choices for the first digit (1, 2, 3, or 4). After selecting the first digit, we have three choices left for the second digit, two choices left for the third digit, and only one choice left for the fourth digit. Therefore, the total number of four-digit numbers that can be formed with distinct digits is 4 x 3 x 2 x 1 = 24.

Explanation
There are 5 students and 5 beds, so each student can be assigned to a specific bed in 5 different ways. Therefore, the total number of ways to allot the beds to the students is 5 x 5 x 5 x 5 x 5 = 125. However, this includes cases where the same bed is assigned to multiple students. Since each bed can only be assigned to one student, we need to divide this total by the number of ways to arrange the students, which is 5! (5 factorial). 125 divided by 5! equals 120.

Explanation
The maximum possible number of 3-letter words in English that do not contain any vowels can be calculated by finding the number of possibilities for each letter position. Since there are 26 letters in the English alphabet, and we are looking for words without vowels, we have 21 consonants to choose from for each position. Therefore, the total number of possibilities is 21 * 21 * 21 = 9261.