I\'m having trouble figuring out how to answer these questions using statistics.

Question

I'm having trouble figuring out how to answer these questions using statistics. If you could show the work behind figuring out these instead of just looking up the answers it would be a huge help!

How many integers from 1000 through 9999 have distinct digits? How many integers from 1000 through 9999 have repeating digits? How many odd integers from 1000 through 9999 have distinct digits? What is the probability that a randomly chosen four-digit integer has distinct digits? What is the probability that a randomly chosen four-digit integer has distinct digits and is odd?

Explanation / Answer

1. To have distinct digits, the digit for the first place can be selected in C[9,1] = 9 ways.(wehave to select any digit between 1 to 9)

Having selected the first digit, the second digit can be selected in C[9,1] = 9 ways (select any number between 0 to 9, except the first number)

Having selected the first and second digits, the third digit can be selected in C[8,1] = 8 ways

Having selected the first, second and third digits, the 4th digit can be selected in C[7,1] = 7 ways

So the number of integers between 1000 to 9999 having distinct digits = 9x9x8x7 = 4536

2. Between 1000 and 9999, there are 9999-999 = 9000 integers

Among these, there are 4536 integers having distinct digits. So the number of integers with repeating digits = 9000-4536 = 4464

3. For the integer to be odd, the last digit should be any one of the digits 1,3,5,7 or 9.

So we can select the last digit in C[5,1] = 5 ways

Then the 1st digit can be selected in C[8,1] = 8 ways

The 2nd digit can be selected in C[8,1] = 8 ways

The 3rd digit can be selected in C[7,1] =7 ways

So the number of odd integers between 1000 through 9999 having distinct digits = 5x8x8x7 = 2240

4. Probability that a randomly chosen 4-digit integer has disctinct digits = 4536/9000 = 0.504

5. Probability that a randomly chosen 4-digit integer has disctinct digits and is odd = 2240/9000 = 0.2489

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