AMC 10 Daily Practice Round 1

Complete problem set with solutions and individual problem pages

Problem 14 Medium

Using numbers: 0, 1, 2, 3, 4, 5, how many four-digit numbers can be formed without repetition, that are greater then 3000, but less than 5421?

  • A.

    180

  • B.

    175

  • C.

    160

  • D.

    100

  • E.

    90

Answer:B

There are four cases:

 

1. When the thousands digit is 3 or 4, the remaining three digits can be chosen freely without repetition:

2 \times 5 \times 4 \times 3 = 120 numbers.

 

2. When the thousands digit is 5 and the hundreds digit is 0, 1, 2, or 3, the remaining two digits can be chosen freely without repetition:

1 \times 4 \times 4 \times 3 = 48 numbers.

 

3. When the thousands digit is 5, the hundreds digit is 4, and the tens digit is 0 or 1, the units digit can be freely chosen from the remaining options:

1 \times 1 \times 2 \times 3 = 6 numbers.

 

4. The number 5420 also satisfies the given conditions.

 

Thus, the total number of valid four-digit numbers is:

 

120 + 48 + 6 + 1 = 175.

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