Question

A four digit number is formed using the digits 0, 1, 2, 3, 4 without repetition. Find the probability that it is divisible by 4.

• A. 3/4
• B. 1/4
• C. 5/16
• D. 7/20

Answer 1

The correct option is C

5/16

We will first find the total number of four digits numbers which can be formed using 0, 1, 2, 3 and 4, without repetition. Let the number be like $\underset{–}{a}\underset{–}{b}\underset{–}{c}\underset{–}{d}$. 'a' can be any of the numbers 1, 2, 3 and or 4. It can't be zero, because then it will become a 3 digit number. Second digit can be filled in 4 ways. [One of the digits of 0, 1, 2, 3 and 4 got used as 'a'. There will be four digits left]. Similarly, c and d can be filled in 3 and 2 ways.

$\Rightarrow$ total number of numbers $=4\times 4\times 3\times 2$

Now we want to find the total number of favorable outcomes. That is, the numbers which are divisible by 4. There can be numbers of the form $\underset{–}{a}\underset{–}{b}12,\underset{–}{a}\underset{–}{b}24,\underset{–}{a}\underset{–}{b}32$, which did not use zero yet and numbers which are of the form $\underset{–}{a}\underset{–}{b}04,\underset{–}{a}\underset{–}{b}20$ and $\underset{–}{a}\underset{–}{b}40$ which has used zero.

In numbers of the form a b 12, we can fill 'a' in 2 ways, and 'b' in 2 ways. So there will be $3\times (2\times 2)$ numbers of that form. Similarly, for numbers of the form a b 04, there can be $3\times (3\times 2)$ ways of filling a and b.

$\Rightarrow$ Total number of favorable outcomes

=Total number of 4-digit numbers divisible by 4

$=3\times (3\times 2)+3\times (2\times 2)$

$=3\times 10=30$

$\Rightarrow$ Probability $=\frac{30}{4\times 4\times 3\times 2}$

$=\frac{5}{16}$