Let S = {2, 3, 4, 5, 6, 7, 9}. How many different 3-digit numbers (with all digits different) from S can be made which are less than 500?
90
Calculating 3-Digit Numbers Less Than 500 from a Given Set
The problem asks us to find the number of different 3-digit numbers that can be formed using the digits from the set S = {2, 3, 4, 5, 6, 7, 9}, with the conditions that all digits must be different and the number must be less than 500.
Let the 3-digit number be represented by H T U, where H is the hundreds digit, T is the tens digit, and U is the units digit.
Applying the Constraints for Forming the 3-Digit Numbers
We have two main constraints:
- All digits (H, T, U) must be different.
- The number H T U must be less than 500.
The second constraint, that the number must be less than 500, directly affects the choice of the hundreds digit (H). For a 3-digit number to be less than 500, the hundreds digit must be less than 5. Looking at the set S = {2, 3, 4, 5, 6, 7, 9}, the digits that are less than 5 are 2, 3, and 4.
- Choosing the Hundreds Digit (H): The possible digits for H are {2, 3, 4}. So, there are 3 choices for the hundreds digit.
Now, let's consider the tens digit (T) and the units digit (U). The digits must be different, and they must come from the set S.
- Choosing the Tens Digit (T): One digit from the set S has already been used for the hundreds place. Since the digits must be different, the tens digit can be any of the remaining digits in S. The set S has 7 digits. After choosing the hundreds digit, there are $7 - 1 = 6$ digits left in S. So, there are 6 choices for the tens digit.
- Choosing the Units Digit (U): Two distinct digits from the set S have now been used (one for the hundreds place and one for the tens place). Since the units digit must also be different from the hundreds and tens digits, it can be any of the remaining digits in S. There are $7 - 2 = 5$ digits left in S. So, there are 5 choices for the units digit.
Calculating the Total Number of Different 3-Digit Numbers
To find the total number of different 3-digit numbers that satisfy all the conditions, we multiply the number of choices for each position (Hundreds, Tens, Units). This is based on the fundamental principle of counting.
Total number of 3-digit numbers = (Choices for H) × (Choices for T) × (Choices for U)
Total number of 3-digit numbers = $3 \times 6 \times 5$
Total number of 3-digit numbers = $18 \times 5$
Total number of 3-digit numbers = $90$
Thus, there are 90 different 3-digit numbers that can be made from the set S = {2, 3, 4, 5, 6, 7, 9} with all digits different and less than 500.
Let's summarize the choices:
| Digit Position | Available Choices (based on constraints) | Number of Choices |
|---|---|---|
| Hundreds (H) | {2, 3, 4} (must be < 500) | 3 |
| Tens (T) | Any remaining digit from S (after choosing H) | 6 |
| Units (U) | Any remaining digit from S (after choosing H and T) | 5 |
Total distinct 3-digit numbers < 500 = $3 \times 6 \times 5 = 90$.
Revision Table: Understanding 3-Digit Number Formation
| Concept | Explanation | Application Here |
|---|---|---|
| Set S | The pool of available digits. | {2, 3, 4, 5, 6, 7, 9} (7 digits) |
| 3-Digit Number | A number with a hundreds, tens, and units place. | HTU format |
| Distinct Digits | Each digit in the number must be unique. | Digits H, T, U must be different. |
| Number < 500 | The value of the number is less than 500. | Hundreds digit (H) must be < 5. |
| Fundamental Counting Principle | If there are $n_1$ ways for task 1, $n_2$ for task 2, etc., total ways are $n_1 \times n_2 \times ...$ | Multiply choices for H, T, U. |
Additional Information: Permutations and Counting Principles
This problem is a classic example of counting principles, specifically related to permutations because the order of the digits matters (e.g., 234 is different from 243). However, a direct permutation formula cannot be applied to the entire set S at once because of the constraint on the hundreds digit.
- Permutation: An arrangement of objects in a specific order. The number of permutations of $n$ distinct objects taken $r$ at a time is denoted by $P(n, r)$ or $_nP_r$ and calculated as $\frac{n!}{(n-r)!}$. In our case, we are selecting 3 digits from a set of 7, but with an additional restriction on the first digit.
- Constraint Handling: When constraints are present (like the hundreds digit being less than 5), it's often easier to handle the constrained positions first. That's why we started by choosing the hundreds digit.
- Step-by-Step Counting: Breaking the problem down into sequential choices (Hundreds, then Tens, then Units) and multiplying the number of options at each step is a powerful technique for solving counting problems with or without constraints.
This method ensures that we count only the numbers that meet all the specified conditions: they are 3-digit numbers, use digits only from set S, have distinct digits, and are less than 500.
Important Questions from Fundamental Principles of Counting
What is the number of four digit decimal number (<1) in which no digit is repeated?
Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four digits appears?
3-digit numbers are formed using the digits 1, 3, 7 without repetition of digits. A number is randomly selected. What is the probability that the number is divisible by 3?
Consider the following paragraph:
THE ABILITY TO REASON ACCURATELY IS VERY IMPORTANT, AS IS THE ABILITY TO COUNT. AS AN EXERCISE IN BOTH, LET US COUNT HOW MANY TIMES THE LETTER "E" OCCURS IN THIS PARAGRAPH. THE CORRECT COUNT IS ________.
Which option when put in the blank in the above paragraph will make the final sentence accurate?
In an examination containing 10 questions, each correct answer is awarded 2 marks, each incorrect answer is awarded −1 and each unattampted question is awarded zero. Which of the following CANNOT be a possible score in the examination?
