Question

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

• A. 2442
• B. 2222
• C. 3333
• D. 2592

Answer 1

The Correct Option is B

To find the arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421,we first need to determine all the possible permutations of the digits.Then,we'll calculate the sum of these permutations and divide by the total number of permutations to find the mean.
The number 1421 has 4 distinct digits:1,2,4, and 2(repeated).
Total number of permutations=\(\frac{4!}{2!}\)(since '2' is repeated)=12
Now, let's list all the distinct permutations:
1241
1214
1421
1412
2141
2114
2411
2411
4121
4112
4211
4211
Now,calculate the sum of these permutations:
Sum=1241+1214+1421+1412+2141+2114+2411+2411+4121+4112+4211+4211=27170
Finally,calculate the mean:

Mean=\(\frac{Sum}{Total\space number\space of\space permutations}\)

= \(\frac{27170}{12}\)≈ 2264.167
The closest option to this mean is option (B).

So,the correct answer is (B): 2222

Answer 2

The number of 4-digit numbers possible using the digits \(1, 1, 2\) and \(4\) \(=\frac {4!}{2!}\)
The count of 1's, 2's, and 4's in the units digit follows a ratio of \(2:1:1\), resulting in 6 occurrences of \(1, 3\) occurrences of \(2,\) and \(3\) occurrences of \(4\).
The sum \(= 6(1)+3(2)+3(4\)
\(=6+6+12\)
\(=24\)
Similarly, this ratio applies to the tens, hundreds, and thousands digits.
Hence, the total sum \(= 24+24(10)+24(100)+24(1000)\)
\(= 24+240+2400+24000\)
\(= 26664\)

Mean \(= \frac {26664}{12}\) = \(= 2222\)

So, the correct option is (B): \(2222\)