How Many Three-Digit Numbers Can Be Formed From the Given Digits with Specific Criteria
This article delves into the problem of determining how many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, and 6, with specific criteria regarding the most significant digit (MSD) and the lowest significant digit (LSD).
Criteria for MSD and LSD
The most significant digit (MSD) of the three-digit number cannot be zero. This leaves us with six choices for the MSD: 1, 2, 3, 4, 5, or 6.
For the lowest significant digit (LSD), the choices are more flexible. However, in even numbers, the LSD can only be one of {0, 2, 4, 6}.
Case 1: Repetition of Digits Allowed
In this case, we first consider the MSD (X), which can be chosen in 6 ways. The LSD (Z) can be any of {0, 2, 4, 6}, so there are 4 choices for Z, and Y can be any of the remaining 7 digits. Therefore, the total number of three-digit numbers is:
6 (choices for X) × 7 (choices for Y) × 4 (choices for Z) 168.
Case 2: No Repetition of Digits
In this scenario, the digits cannot be repeated, and the choices for the MSD (X) and LSD (Z) must be considered carefully.
Without Zero as the MSD
Here, Z can only be 0, 2, 4, or 6. The case when Z is 0 is treated separately:
If Z 0, then there are 6 choices for the middle digit Y (since Y cannot be 0) and 5 choices for the most significant digit X (since X cannot be 0 or the same as Y).The number of three-digit numbers in this case is:
5 (choices for X) × 6 (choices for Y) 30.
Excluding Zero from Choices for Z
When Z is not 0, there are 3 choices for Z from {2, 4, 6}. Y can be any of the remaining 5 digits, and X can be any of the remaining 5 digits except 0 and Y.
The number of three-digit numbers in this case is:
5 (choices for X) × 5 (choices for Y) × 3 (choices for Z) 75.
The total number of three-digit numbers without repetition and without zero as the MSD is:
30 (from Z 0) 75 (from Z ≠ 0) 105.
Combination of Digits Without Restrictions
Without any restrictions on the use of digits, if repetition is allowed, the total number of three-digit numbers is:
7 (choices for X) × 7 (choices for Y) × 7 (choices for Z) 343.
If repetition is not allowed and 0 cannot start the number, the total number of three-digit numbers is:
6 (choices for X) × 6 (choices for Y) × 5 (choices for Z) 180.
Conclusion
The number of three-digit numbers that can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 depends on several factors, including the possibility of repetition and the restriction on the most significant digit being non-zero. The possible counts are:
343 (with repetition allowed) 294 (without repetition and zero as MSD) 210 (without repetition but with zero as MSD) 180 (without repetition and zero not as MSD)The specific count will vary based on the problem's constraints and the author's intent.