Question

A five digit number is formed by the digits 1, 2, 3, 4, 5 with no digit repeated. The probability that the number is divisible by 4 is:
(a) $\dfrac{1}{5}$
(b) $\dfrac{2}{5}$
(c) $\dfrac{3}{5}$
(d) $\dfrac{4}{5}$

Answer 1

Hint: A number is divisible by 4 when its last two digits are divisible by 4 so in the 5 digit number we are looking for the last two digits which are divisible by 4. The probability is equal to the division of favorable outcomes with the total outcomes. Total outcomes are the possible ways without the condition that the number is divisible by 4 and favorable outcomes are the number of possible ways such that the number is divisible by 4.

Complete step-by-step answer:
The probability of any outcomes is equal to:
$\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}$
In this problem, the total outcomes are all the possible ways to make a 5 digit number without the condition that the number is divisible by 4.
To construct a 5 digit number, we are showing the 5 blanks below:

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In the first blank from the 5 numbers any number can appear so 5 possibilities are there for the first blank and as it has said that no number should be repeated so now, one number is gone so for the second blank only 4 possibilities are there. Similarly for third blank, fourth blank and fifth blank has 3, 2 and 1 possibilities respectively. Now, the possibilities in each blank are independent to each other so we multiply the possibilities of each blank.
Hence, the total number of ways to make a 5 digit number so that no number is repeated is:
$\begin{aligned} & 5.4.3.2.1 \\\ & =120 \\\ \end{aligned}$
We can write the above multiplication of numbers as 5!
From the above, the total outcomes are 5! Or 120.
Now, we have to find the probability of a 5 digit number which is divisible by 4.
We know that the divisibility rule for 4 is that the last two digits of the number must be divisible by 4.
To make a 5 digit number we have shown 5 blanks in the below:

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Now, in the above 5 blanks there is no restriction on first 3 digits because divisibility rule of 4 says that last two numbers must be divisible by 4. For a number to be divisible by 4 first of all it should be an even number and for a number to be an even number the last digit of the number must be divisible by 2 so from the 5 digit numbers 1, 2, 3, 4, 5; 2 or 4 should be at the end of the number.
When 2 is the last digit then with 2 as the second digit then the first digit could be 1, 3, 5 so that the last two digits of the number are divisible by 4. Now, we have fixed the last digit as 2 so the last blank contains only 1 possibility then the last second blank contains 3 possibilities (1, 3, 5). In the following we have shown three ways in which the last two digits are divisible by 4.
_ _ _ 12, _ _ _ 32, _ _ _ 52
Now 2 numbers are gone in the last two blanks so we are left with 3 numbers and 3 blanks so arranging 3 numbers in the 3 blanks is 3!
Then the possible 5 digit numbers are:
$\begin{aligned} & \left( 3! \right)\left( 3 \right)\left( 1 \right) \\\ & =6\left( 3 \right) \\\ & =18 \\\ \end{aligned}$
Similarly, fixing 4 in the last digit of 5 digit number the possible numbers which are divisible by 4 are:
_ _ _ 24
From the above, only 1 possibility is there when fixing 4 as the last digit such that the last two digits of the number should be divisible by 4. Now, 2 numbers are gone i.e. 2 and 4 and we are left with 3 numbers so we have 3 blanks and 3 numbers so arranging 3 numbers in 3 blanks we have 3!
Hence, when 4 is at the last digit then the number of possible 5 digit numbers are:
$\begin{aligned} & 3! \\\ & =3.2.1 \\\ & =6 \\\ \end{aligned}$
Total possible 5 digit numbers which are divisible by 4 is:
$\begin{aligned} & 18+6 \\\ & =24 \\\ \end{aligned}$
From the above, favorable outcomes are 24 and total outcomes are 120. So, the probability is equal to:
$\begin{aligned} & \dfrac{\text{Favorable outcomes}}{\text{Total outcomes}} \\\ & =\dfrac{24}{120} \\\ & =\dfrac{1}{5} \\\ \end{aligned}$
Hence, the correct option is (a).

Note: Be careful while finding the 5 digit number without applying the condition that the number should be divisible by 4 is that it is given that the numbers should not be repeated means any of the digits in 5 digit number should not be repeated. Generally, it has been observed that in the hastiness of solving the problem, students ignore this condition. If you miss this statement then the whole problem will go in different directions.
The number of possible ways of writing 5 digit numbers so that digits are repeated then each of the 5 blanks can get any number from 1 to 5 so the possible ways are:
${{\left( 5 \right)}^{5}}$
So, you can see that the answer is completely different from what we have got above.