Question

Probability of all $4$–digit numbers having all the digits same is:

• A. $\frac{1}{4}$
• B. $\frac{9}{1000}$
• C. $\frac{1}{1000}$
• D. $\frac{1}{100}$

Answer 1

The correct option is C

$\frac{1}{1000}$

Explanation of the correct option:

Find the required probability:

Total no. of digits $=10$ $\left(0\mathrm{to}9\right)$

For forming the four digit number using these digits,

The thousands place can be filled in $9$ ways. (excluding $0$)

Hundreds, tens and ones place can be filled in $10$ ways each, as repetition is allowed.

$\begin{array}{rcl}& \Rightarrow & n\left(S\right)=9\times 10\times 10\times 10\\ & =& 9000\end{array}$

Let $A$ be the event that in the four digit number formed by using the given digits, all the digits are same.

The thousands place can be filled in $9$ ways. (excluding $0$)

Hundreds, tens and ones place can be filled in $1$ways each, as all the digits are same.

$\begin{array}{rcl}& \Rightarrow & n\left(A\right)=9\times 1\times 1\times 1\\ & =& 9\end{array}$

Therefore, the required probability:

$\begin{array}{rcl}P\left(A\right)& =& \frac{n\left(A\right)}{n\left(S\right)}\\ & =& \frac{9}{9000}\\ & =& \frac{1}{1000}\end{array}$

Hence, the correct answer is option C.