Question

A six digit number is to be formed from the given numbers 1 2 3456.Find the probability that number is divisible by 4

Answer 1

Here's no condition that the digit repetition is not occurred! So digit repetition can be occurred!

Given digits are 1, 2, 3, 4, 5 and 6. No. of digits is 6.

Thus no. of six digit numbers formed by these numbers is 6⁶ = 46656.

This is the total no. of outcomes.

We know that a number is divisible by 4 if the no. formed by the last two digits is also divisible by 4.

When we make six digit numbers from the given numbers 1, 2, 3, 4, 5 and 6 which are divisible by 4, then they will end in either the following two digit numbers:

12, 16, 24, 32, 36, 44, 52, 56, 64

Let each two digit numbers be a one-digit number or a set and let this set be indicated as X. Thus, as an example, a six digit number 111112 which can be formed and is divisible by 4, can be written as 1111X. [X is put at the place of 12]

So the possible values for X are mentioned above. No. of these possible values is 9.

As digit repetition can be occurred, the no. of six digit numbers ending in each value of X would be same.

Consider the row given below.

$\begin{tabular}{|c|c|c|c|c|}\cline{1-5}&&&&X\\ \cline{1-5}\end{tabular}$

This is a space entering for a six digit number, where the last digit represented by X is actually of two digit space.

Since digit repetition is occurred, the first 4 spaces can be given by the all given numbers 1, 2, 3, 4, 5, 6. So 6 numbers can be entered there.

And the last space, represented by X, can be given by all values of X. So 9 numbers can be entered there.

$\begin{tabular}{|c|c|c|c|c|}\cline{1-5} \ & \ \ & \ \ & \ \ &X\\ \cline{1-5}\end{tabular}\\ \begin{tabular}{ccccc}6&6&6&\ 6&\ 9\end{tabular}$

Thus the total no. of six digit multiples thus can be formed = 6 × 6 × 6 × 6 × 9 = 6⁴ × 9 = 11664

This is the no. of favourable outcomes.

So, probability,

$\displaystyle \Longrightarrow\ \frac{11664}{46656} \\ \\ \\ \\ \Longrightarrow\ \frac{6^4 \times 9}{6^6} \\ \\ \\ \\ \Longrightarrow\ \frac{9}{6^2} \\ \\ \\ \\ \Longrightarrow\ \frac{9}{36} \\ \\ \\ \\ \Longrightarrow\ \Large \text{ \bold{\frac{1}{4}} }$

Thus the answer is 1/4.