Question

How many numbers greater than 24000 and less than 54000 can be formed using digits 0 1 2 3 4 5 6?

Answer 1

First we find the number of 5 digit integers greater than 24000 that can be formed using the digits 0,1,2,3,4,5,6 repetitions allowed.

The possible integers are the sum of those starting with 2 but greater than 24000 and those that start with 3,4,5,6. That is

$3*7^3-1+4*7^4=10632$.

Next we find the number of 5 digit integers greater than 54000 that can be formed using the digits 0,1,2,3,4,5,6 repetitions allowed.

The possible integers are the sum of those starting with 5 but greater than 54000 and those that start with 6. That is

$3*7^3-1+1*7^4=3429$.

Now we can write

[tex]N(24000<X<54000)=N(X>24000)-N(X>54000)-1\\ N(24000<X<54000)=10632-3429-1\\ N(24000<X<54000)=7,202[/tex]

Answer 2

answer:-

4 choices for the first digit (one of 3, 4, 5, 6) for 5-digit numbers greater than 30,000.

6 choices for the second digit (one of 0, 1, 2 and one of ( 3, 4, 5, 6 minus the 1st digit),

5 choices for the third digit, (any one of the remaining 5 digits)

4 choices for the fourth digit, (any one of the remaining 4 digits)

3 choices for the fifth digit. (any one of the remaining 3 digits)

Therefore,

4 * 6 * 5 * 4 * 3 = 1,440 5-digit numbers formed greater than 30,000 and therefore greater than 24,000.

For 5-digit numbers starting with 2 and greater than 24,000, we have:

3 choices for the 2nd digit (one of 4, 5, 6),

5 choices for the 3rd digit (one of 0, 1, 3) and one of (4, 5, 6, minus digit used for 2nd digit),

4 choices for the 4th digit (any one of the remaining 4 digits)

3 choices for the 5th digit (any one of the remaining 3 digits)

Therefore,

1 * 3 * 5 * 4 * 3 = 180 5-digit numbers formed greater than 24,000 starting with 2.

Therefore,

1,440 + 180 = 1,620 5-digit numbers greater than 24,000 can be formed from digits 0, 1, 2, 3, 4, 5, 6 when repetition is not allowed. The smallest is 24,013, and the largest is 65,432.