Question

What is the average of all possible five-digit numbers that can be formed by using
each of the digits 6, 7,5,9, and 2 exactly once?​

Answer 1

Given:

The digits 6, 7, 5, 9, and 2.

5 digit numbers are to formed using the above digits without repetition.

To find:

The average of all possible five digit numbers.

Solution:

First of all, let us have a look at the total count of number that can be formed using these five digits.

The repetition is not allowed so total number of possibilities are:

$5\times 4\times3\times2\times1=120$

Now, it can also be observed that each digit is not repeated and all are unique therefore, each digit appears exactly $\frac{120}{5} =24$ number of times at each place i.e. one's, ten's, hundred's, thousand's and ten thousand's.

Sum of all the digits = $(6+7+5+9+2)\times 24 = 29\times 24=696$

Sum of all the numbers such formed = $696 \times (10^4+10^3+10^2+10^1+1) = 696\times 11111 = 7733256$

$Average = \dfrac{\text{Sum of all the numbers}}{\text{Total count of numbers}}$

The required average will be:

$Average = \dfrac{7733256}{120} = \bold{64443.8}$