Identify how many digits the possible integers must have. Which integers can be used in the first position?
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Practice makes perfect
We are asked how many integers greater than 999 but not greater than 4000 can be made with the digits 0, 1, 2, 3, and 4. This means that every possible integer is between 1000 and 4000. Because of this, the integer must have four digits.
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Using the Fundamental Counting Principle, we will multiply the ways in which we can assign each digit to find the possible integers.
( Choices for 1st digit ) * ( Choices for 2nd digit ) *
( Choices for 3rd digit ) * ( Choices for 4th digit )
Let's consider the first digit on the left. Since the numbers are greater than 999, the first digit cannot be 0. Also, let's only consider the numbers less than 4000 because this number is the upper limit. Doing so, we know that there are 3 choices for the first digit — 1, 2, and 3.
( 3 ) * ( Choices for 2nd digit ) *
( Choices for 3rd digit ) * ( Choices for 4th digit )
Every four digit number that starts in 1, 2, or 3 is a number in the desired interval. Therefore, the choices for the other digits are not restricted. Since we can repeat digits, the number of choices for the other digits is always 5 — from 0 to 4.
( Choices for 1st digit ) * ( Choices for 2nd digit ) * ( Choices for 3rd digit ) * ( Choices for 4th digit )
