Three digit numbers Trivia Quiz

Answer: 900

There are 9 possibilities for the first digit, 10 for the second, and 10 for the third. Thus there are 9 * 10 * 10 = 900 different three digit numbers.

Answer: 450

A three digit number is even if the last digit is 0, 2, 4, 6, or 8. There are 9 possibilities for the first digit, 10 for the second, and 5 for the third. Hence there are 9 * 10 * 5 = 450 different even three digit numbers.

Answer: 648

There are 9 possibilities for the first digit. There are also 9 possibilities for the second digit since 0 can be used. The second digit cannot equal the first digit. The third digit cannot equal either the first or the second, so there are 8 possibilities for it. Thus there are 9 * 9 * 8 = 648 three digit numbers with no repeated digits.

Answer: 252

By question one, there are 900 three digit numbers. By question three, 648 of them have no repeated digits. Since a three digit number either has no repeated digits or at least one repeated digit, the number of three digit numbers having at least one repeated digit is 900 - 648 = 252.

Answer: 171

A three digit number either doesn't contain the digit zero or it contains at least one zero. The number of three digit numbers that do not contain zero is 9 * 9 * 9 = 729 (9 possibilities for each digit). By question one, there are a total of 900 three digit numbers. Hence there are 900 - 729 = 171 three digit numbers that contain the digit 0.

Answer: 128

1000 divided by 7 is 142 with remainder 6. So there are 142 positive integers less than 1000 that are divisible by 7. 100 divided by 7 is 14 with remainder 2. So there are 14 positive integers less than 100 that are divisible by 7. Hence there are 142 - 14 = 128 three digit numbers that are divisible by 7.

Answer: 22

The square root of 1000 is 31.6 to one decimal place. So there are 31 perfect squares less than 1000. 10 squared is 100, so there are 9 perfect squares less than 100. Hence there are 31 - 9 = 22 three digit numbers that are perfect squares. The first one is 100, the last is 961.

Answer: 84

First of all note that 0 cannot be a digit. Using my hint, the answer is 9 * 8 * 7/ 6 = 84. 6 is the number of times each three digit number is counted if order of the digits doesn't matter. 9 * 8 * 7 is the number of three digit numbers that don't contain 0 whose digits don't repeat. Here is another solution if you know about combinations:

Consider the set {1, 2, ..., 9}. Any subset of three elements of this set would be the digits of a three digit number. But there is only one arrangement of the three digits that is in increasing order. So a three element subset determines a unique three digit number with the digits in increasing order. Therefore, the number of such three digit numbers is equal to the number of three element subsets of this nine element set, which is just the number of combinations of 9 objects, taken 3 at a time: C(9, 3) = 9!/(3! * 6!) = 84. An equivalent way,

Answer: 120

In this case 0 can be a digit. Using the same reasoning as in question 8, we get (10 * 9 * 8)/6 = 120 such three digit numbers. Here is another solution involving combinations:

As in the solution to question eight, the number of such three digit numbers is the same as the number of subsets of the ten element set {0, 1, ..., 9}, which is C(10, 3) = 120.

Answer: 15

Note that 0 cannot be the first digit. We can list all the possibilities:

First digit 1: 104, 113, 122, 131, 140
First digit 2: 203, 212, 221, 230
First digit 3: 302, 311, 320
First digit 4: 401, 410
First digit 5: 500

Hence there are 15 three digit numbers whose digits sum to 5. I hope you enjoyed my quiz! Thanks for playing!