Question

Three digit numbers are formed using the digits 0, 2, 4, 6, 8. A number is chosen at random out of these numbers. What is the probability that this number has the same digits?

(a) $\frac{1}{16}$ (b) $\frac{16}{25}$ (c) $\frac{1}{645}$ (d) $\frac{1}{25}$

Answer 1

The given digits are 0, 2, 4, 6, 8.

___ ___ ___
Hundreds Tens Ones

Now, there are 4 ways to fill the hundreds place (0 cannot occupy the hundreds place),
There are 5 ways to fill the tens place and 5 ways to fill the ones place.
∴Total number of 3 digits = 4 × 5 × 5 = 100
The three digit numbers formed using same digits are 222, 444, 666 and 888.
i.e. Number of 3 digit numbers that have the same digits = 4
∴ Probability of three digit number formed with same digits = $\frac{\mathrm{number}\mathrm{of}3\mathrm{digit}\mathrm{number}\mathrm{have}\mathrm{same}\mathrm{digits}}{\mathrm{total}\mathrm{number}\mathrm{of}3\mathrm{digit}\mathrm{number}}$
= $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$100$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.$
Hence, the correct answer is option D.