Math Probability homework :(

Question

hello! im an eight grade student and im having trouble with my online seatwork. Can someone help me?

1. License plates of automobiles consist of 3 letters of the English alphabet and a three-digit number. How many different license plates are possible if no number or letter is repeated?
2. A class consists of 15 boys and 21 girls. If all the class members can sing, how many different duets can be formed with a boy and a girl combinations?
3. An ATM pin consists of 4 digits. How many different pins can be made if the digit can repeat?
4. How many even numbers between 500 and 700 can be formed from the digits 3, 4, 6, 7, 8, 9, without repetition?
5. There are 10 finalists in a beauty pageant. In how many ways can a Beauty Queen, a First Runner-up, and Second Runner-up be awarded?
6. A True or False test consist of 10 questions. If all the questions are answered, how many different ways can this be done?
7. In a 6/42 lotto game, how many different bets are possible?
8. A landline telephone number has 7 digits. How many different landline numbers are possible if it must not start with zero? (with repetition)
9. A 10-item multiple choice test has 3 choices for each question. In how many ways can the test be answered completely?
10. An organization has 10 members who are qualified to be elected in the positions president, vice president, secretary, treasurer, and auditor. In how many ways can the officers be formed?

i need the answers asap, the due date is today :((

DEADLINE 2:00 pm Tuesday, March 16, 2021 (GMT+8)

Answer 1

1). License plate. 3 letters, 3 numbers. order matters, values can repeat, use power. 26³ * 10³ = 17576*100 or 1,756,600 combinations.

2) For every girl, she can have 15 partners. 21 girls x 15 = 315 possible pairs

3) If numbers can repeat, total of 4 10 digit combinations is 10⁴ or 10,000 (including 0000)

4) 7, 8, and 9 can not be used for the first number because the number can not exceed 700. 3 and 4 can not be used for the first number because the whole number must be larger than 500. So The first number must be 6.

Even numbers must end in an even number. That gives 2 numbers for the last digit (4,8)

The middle number can be 3,7,9,8, or 4, however 8 and 4 can only be used half the time each because when the last number is 8 , it can not be in the middle and when 4 is the last number it can not be used in th middle. Total number of combinations with the first digit fixed at 3 and the last digit either 4 or 8 would be 4*2 or 8,

5). -9 --- you should be able to start seeing some patterns here I leave this for the student.

10) this is a combination problem where order does matter since the position of president is not the same as the position of auditor. ₁₀P₅. or 30,240