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Question number 1: By counting the right sides vertical lengths you get a total of 10. On the left side you get 5 + x. So, 5+x=10, x = 5.
Question number 2: The first number has 9 possibilities, 1-9. The second number has 9 possibilities as it is 0-9, but not the first number. The third number has 8 possibilities, 0-9, but not the first two numbers. The last digit has 7 possibilities, 0-9, but not the other three numbers. So, 9*9*8*7 = 4536.
Question number 3: Let M be the midpoint of AB. The area of the triangle can be found by base AB and height MC, because the triangle is isosceles. Using Pythagorean theorem AB = 2sqrt5, and MC =
sqrt5. Then find the area by 1/2bh. (2sqrt*sqrt5)/2 = 5. The grid’s area is 30, so 5/30 = 1/6.
Question number 4: The first place is one of 5 letters. The second place is one of 21 letters, which means the third place is one of 20 letters. The last place is one of 10 numbers. This is very straightforward, so 5*21*20*10 = 21000 are the total number of solutions. AMC8 being one of the possibilities has a 1/21000 chance of being made.