The discussion revolves around finding the number of distinct 4-digit numbers greater than 3000 that can be formed using the digits 2, 2, 3, 3, 3, 4, 4, 4, 4. Participants explore different approaches to solve the problem, including case analysis based on the first digit.
There is no consensus on the correct method or total number of distinct 4-digit numbers, as one participant suggests a total of 54, while others arrive at 51 through case analysis.
The discussion includes various assumptions about the selection of digits and the conditions under which certain combinations are valid, which remain unresolved.
2×3×3×3=54, is this correct sir? I am confused since 51 is the key answerMarkFL said:What have you tried, and where are you stuck?
Thank you sirMarkFL said:The first digit is either 3 or 4. So, let's consider each case separately:
(1) First digit is 3:
Then the rest of the numbers must come from the list: 2, 2, 3, 3, 4, 4, 4, 4
Therefore we may choose any 3-digit sequence except 222 and 333 for the rest of the digits. This shows there are:
$$N_1=3^3-2=25$$
numbers in this case.
(2) First digit is 4:
Then the rest of the numbers must come from the list 2, 2, 3, 3, 3, 4, 4, 4
Therefore we may choose any 3-digit sequence except 222 for the rest of the digits. This shows there are:
$$N_2=3^3-1=26$$
numbers in this case.
Hence, the total number is:
$$N=N_1+N_2=51$$