The arithmetic mean of the following numbers 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6 and 7, 7, 7, 7, 7, 7, 7, is

The correct option is 5.

To find the arithmetic mean of a set of numbers, we divide the sum of all the numbers by the total count of numbers in the set. Let's write out the given dataset and analyze the frequency of each number:

The number 1 appears 1 time.
The number 2 appears 2 times.
The number 3 appears 3 times.
The number 4 appears 4 times.
The number 5 appears 5 times.
The number 6 appears 6 times.
The number 7 appears 7 times.

First, we calculate the total count of numbers ($N$) by summing the frequencies of each number:

$N=1+2+3+4+5+6+7$

Using the formula for the sum of the first $n$ natural numbers, $S=\frac{n(n+1)}{2}$ where $n=7$:
$N=\frac{7\times 8}{2}=28$

Next, we calculate the sum of all the numbers in the dataset. Each number $i$ (from 1 to 7) appears $i$ times, so the sum of each number's occurrences is $i^2$. The total sum ($S$) is given by:

$S=(1\times 1)+(2\times 2)+(3\times 3)+(4\times 4)+(5\times 5)+(6\times 6)+(7\times 7)$

$S=1^2+2^2+3^2+4^2+5^2+6^2+7^2$

Using the formula for the sum of squares of the first $n$ natural numbers, $S=\frac{n(n+1)(2n+1)}{6}$ where $n=7$:
$S=\frac{7\times 8\times 15}{6}$

Simplifying the expression:
$S=\frac{840}{6}=140$

Finally, we compute the arithmetic mean ($\mu$) by dividing the sum of the numbers by the total count:

$\mu =\frac{S}{N}=\frac{140}{28}$

$\mu =5$

Thus, the arithmetic mean of the given numbers is 5.