Question

How many three-digit numbers can be formed using the digits 0 and 1 ? Repeated digits are allowed.

Short answer

4

Step-by-step solution

Understand the problem

We need to find how many three-digit numbers can be formed using the digits 0 and 1, with repetition allowed.

Identify the valid digits for each position

For a three-digit number, the hundreds place cannot be 0 (otherwise it would not be a three-digit number). Therefore, the hundreds place can only be 1.

Determine the choices for each position

Since the hundreds place must be 1, we only need to decide the digits for the tens and ones places. Both can be either 0 or 1.

Calculate the total number of combinations

The hundreds place has 1 choice (1). The tens place has 2 choices (0 or 1), and the ones place also has 2 choices (0 or 1). The total number of combinations is calculated by multiplying these choices: \[ 1 \times 2 \times 2 = 4 \]

Key concepts

concept of combinatorics

Combinatorics, in simple terms, is the branch of mathematics that deals with counting. It helps us figure out how many ways we can arrange or choose items. In this exercise, we are dealing with how many three-digit numbers can be formed using 0 and 1. To solve this, we are essentially counting the different possible combinations of 0s and 1s in each digit place of a three-digit number. When repetition of digits is allowed, this task becomes easier because each position (place value) has the same set of options to choose from, except for the hundreds place since a three-digit number must start with a non-zero digit.

repetition of digits

Repeating digits means you can use the same digits more than once. In our problem, we are allowed to use both 0 and 1 repeatedly in the tens and ones places. This means we can have combinations like 100, 101, 110, and 111.
Let's break it down:

• The hundreds place must be 1 (since it cannot be 0).
• The tens place can be either 0 or 1, giving us 2 choices.
• The ones place also can be either 0 or 1, providing another 2 choices.

Repetition simplifies our counting process as each position is independent of the other, except for any constraints like the one in the hundreds place.

place value

Place value refers to the position of a digit in a number which determines its value. In our three-digit number, we have three places: the hundreds place, the tens place, and the ones place.
Important to note:

• The hundreds place is the leftmost digit in a three-digit number and must be non-zero, as it represents values from 100 to 999.
• The tens place is one step to the right of the hundreds place, contributing tens to the overall value.
• The ones place is the rightmost position, contributing the least but equally significant in counting total digit combinations.

For example, in 101, the 1 in the hundreds place means 100, the 0 in the tens place means 0, and the 1 in the ones place means 1. Together, they form the number 101. Understanding place value helps us determine how each digit affects the overall number, especially in combinatorial problems like the one in this exercise.