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Quiz about Three digit numbers
Quiz about Three digit numbers

Three digit numbers Trivia Quiz


In each question, you are asked to count the number of three digit numbers having a certain property. Note that zero cannot be the first digit. These problems are combinatorial in nature and can be solved mathematically without guessing. Good luck!

A multiple-choice quiz by rodney_indy. Estimated time: 10 mins.
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Author
rodney_indy
Time
10 mins
Type
Multiple Choice
Quiz #
267,756
Updated
Jul 23 22
# Qns
10
Difficulty
Difficult
Avg Score
4 / 10
Plays
1143
Awards
Top 20% Quiz
- -
Question 1 of 10
1. The multiplication principle is an important method of counting. For example, suppose we had a contest that awarded a first prize, a second prize, and a third prize. Suppose also nobody could win more than one prize. If there were 15 people in the contest, then there would be 15 who could win first prize, 14 who could win second, and 13 who could win third. The total number of ways the prizes could be awarded would be 15 * 14 * 13 = 2730 ways.

You can use the multiplication principle to answer the following question:

How many three digit numbers are there? Remember that the first digit cannot be 0.

Answer: (Find how many possibilities there are for each digit, then multiply)
Question 2 of 10
2. How many three digit numbers are even?

[Once again, we can use the multiplication principle. Count the number of possibilities for each digit, but keep in mind that an even number can only end in certain digits.]

Answer: (Find how many possibilities there are for each digit, then multiply)
Question 3 of 10
3. How many three digit numbers have no repeated digits?

[Be careful with this one. The second digit cannot be the same as the first digit, but 0 can be used as the second digit. The third digit cannot be the same as either of the first two digits.]

Answer: (Find how many possibilities there are for each digit, then multiply)
Question 4 of 10
4. How many three digit numbers have at least one repeated digit?

[For this problem, note that if we added (the number of three digit numbers that had no repeated digits) to (the number of three digit numbers with at least one repeated digit), we would obtain (the total number of three digit numbers). Two of the quantities in parentheses have already been computed.]

Answer: (subtract two numbers)
Question 5 of 10
5. How many three digit numbers contain the digit 0?

[For this problem, let's try a procedure analagous to the one we used in question 4. Note that (the number of three digit numbers that don't contain the digit 0) + (the number of three digit numbers that contain 0) = (the total number of three digit numbers).]

Answer: (subtract two numbers)
Question 6 of 10
6. How many three digit numbers are divisible by 7?

[This problem is different. For example, to find the number of positive integers less than 1000 that are divisible by 17, divide 1000 by 17. You will get a quotient of 58 and a remainder of 14. This means that there are 58 positive integers that are less than 1000 that are divisible by 17. But if you want the number of three digit numbers divisible by 17, you would also have to find the number of positive integers less than 100 that are divisible by 17 by division.]

Answer: (Subtract two numbers)
Question 7 of 10
7. How many three digit numbers are perfect squares?

[For this one you will need to look at square roots. Don't count the numbers less than 100.]

Answer: (Subtract two numbers)
Question 8 of 10
8. How many three digit numbers have their digits in increasing order (such as 358)? Note also that the digits cannot be repeated.

[This one is a little tricky. Note that there are 3 * 2 * 1 = 6 ways the digits of any particular three digit number can be arranged to form a three digit number with the same digits. For example, the digits in the number 385 can be rearranged to form a total of six different three digit numbers (including 385 itself): 385, 358, 538, 583, 835, 853. Only one of these ways, 358, has the digits in increasing order. So if we counted all the three digit numbers with no digits that repeat that don't contain zero, we would get six times the answer to the question.]

Answer: (Not a permutation)
Question 9 of 10
9. How many three digit numbers have their digits in decreasing order (such as 820)? Note also that the digits cannot be repeated.

[This one is similar to the last question, but it has a different answer since 0 may appear in the number.]

Answer: (Not a permutation)
Question 10 of 10
10. Finally, how many three digit numbers are there whose digits sum to 5?

[For this problem, you can list all the possibilities rather quickly by the following method. First of all, the first digit can be 1, 2, 3, 4, or 5. Now if the first digit is 1, the second digit can be 0, 1, 2, 3, or 4. Keep going, you'll quickly come up with all of them.]

Answer: (They can be enumerated logically.)

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Quiz Answer Key and Fun Facts
1. The multiplication principle is an important method of counting. For example, suppose we had a contest that awarded a first prize, a second prize, and a third prize. Suppose also nobody could win more than one prize. If there were 15 people in the contest, then there would be 15 who could win first prize, 14 who could win second, and 13 who could win third. The total number of ways the prizes could be awarded would be 15 * 14 * 13 = 2730 ways. You can use the multiplication principle to answer the following question: How many three digit numbers are there? Remember that the first digit cannot be 0.

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.
2. How many three digit numbers are even? [Once again, we can use the multiplication principle. Count the number of possibilities for each digit, but keep in mind that an even number can only end in certain digits.]

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.
3. How many three digit numbers have no repeated digits? [Be careful with this one. The second digit cannot be the same as the first digit, but 0 can be used as the second digit. The third digit cannot be the same as either of the first two digits.]

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.
4. How many three digit numbers have at least one repeated digit? [For this problem, note that if we added (the number of three digit numbers that had no repeated digits) to (the number of three digit numbers with at least one repeated digit), we would obtain (the total number of three digit numbers). Two of the quantities in parentheses have already been computed.]

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.
5. How many three digit numbers contain the digit 0? [For this problem, let's try a procedure analagous to the one we used in question 4. Note that (the number of three digit numbers that don't contain the digit 0) + (the number of three digit numbers that contain 0) = (the total number of three digit numbers).]

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.
6. How many three digit numbers are divisible by 7? [This problem is different. For example, to find the number of positive integers less than 1000 that are divisible by 17, divide 1000 by 17. You will get a quotient of 58 and a remainder of 14. This means that there are 58 positive integers that are less than 1000 that are divisible by 17. But if you want the number of three digit numbers divisible by 17, you would also have to find the number of positive integers less than 100 that are divisible by 17 by division.]

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.
7. How many three digit numbers are perfect squares? [For this one you will need to look at square roots. Don't count the numbers less than 100.]

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.
8. How many three digit numbers have their digits in increasing order (such as 358)? Note also that the digits cannot be repeated. [This one is a little tricky. Note that there are 3 * 2 * 1 = 6 ways the digits of any particular three digit number can be arranged to form a three digit number with the same digits. For example, the digits in the number 385 can be rearranged to form a total of six different three digit numbers (including 385 itself): 385, 358, 538, 583, 835, 853. Only one of these ways, 358, has the digits in increasing order. So if we counted all the three digit numbers with no digits that repeat that don't contain zero, we would get six times the answer to the question.]

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,
9. How many three digit numbers have their digits in decreasing order (such as 820)? Note also that the digits cannot be repeated. [This one is similar to the last question, but it has a different answer since 0 may appear in the number.]

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.
10. Finally, how many three digit numbers are there whose digits sum to 5? [For this problem, you can list all the possibilities rather quickly by the following method. First of all, the first digit can be 1, 2, 3, 4, or 5. Now if the first digit is 1, the second digit can be 0, 1, 2, 3, or 4. Keep going, you'll quickly come up with all of them.]

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!
Source: Author rodney_indy

This quiz was reviewed by FunTrivia editor crisw before going online.
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