Question

The number of positive integers less than 1000 having only odd digits is

• A. 155
• B. 177
• C. 55
• D. 205
• E. 85

Hint:

Combinatorics Tip: "Less than $10^n$" is a classic combinatorics phrase meaning you must account for numbers of lengths 1 up to $n$. Don't just calculate the 3-digit numbers and stop!

Answer 1

The Correct Option is A

Concept:
To find the total number of integers under 1000 fitting this condition, we must break the problem down by digit length (1-digit, 2-digit, and 3-digit numbers) and use the Fundamental Principle of Counting for each category.

Step 1: Identify the available digits.
The problem states the numbers must have only odd digits. The set of odd digits is $\{1, 3, 5, 7, 9\}$. This gives us exactly 5 available choices for any single digit position.

Step 2: Count the 1-digit numbers.
A 1-digit number has only one slot to fill. Number of ways to fill it = 5 choices. Total 1-digit numbers = 5

Step 3: Count the 2-digit numbers.
A 2-digit number has two slots (tens, ones). Choices for the tens digit = 5 Choices for the ones digit = 5 Using the multiplication principle: $5 \times 5 = 25$ Total 2-digit numbers = 25

Step 4: Count the 3-digit numbers.
A 3-digit number has three slots (hundreds, tens, ones). Choices for the hundreds digit = 5 Choices for the tens digit = 5 Choices for the ones digit = 5 Using the multiplication principle: $5 \times 5 \times 5 = 125$ Total 3-digit numbers = 125

Step 5: Calculate the grand total.
Since an integer less than 1000 must be either a 1-digit, 2-digit, OR 3-digit number, we sum the combinations: $$\text{Total} = 5 + 25 + 125$$ $$\text{Total} = 155$$ Hence the correct answer is (A) 155.