Question

Write the sample space $S$ and $n\left( S \right)$ for ‘making a two-digit number using the digits 0, 1, 2, 3; without repetition’.

Answer 1

Here, we will find the answer using three cases. The digit at tens place of a two-digit number cannot be 0. The three cases are – the digit at the tens place is 1, 2 or 3. Then we will put the rest of the numbers one by one in the units place to find the sample space.

Complete step by step solution:
We know that the digit at tens place of a two-digit number cannot be 0. Thus, the digit at tens place can either be 1, 2, or 3.
Case 1: The digit at tens place is 1.
Now, we have used the digit 1 at the tens place.
The remaining digits are 0, 2, and 3.We will put the remaining digits 0, 2, 3 one by one in the unit’s place to find three two-digit numbers.
Thus, we get the two-digit numbers 10, 12, and 13.
Case 2: The digit at tens place is 2.
Now, we have used the digit 2 at the tens place.
The remaining digits are 0, 1, and 3.We will put the remaining digits 0, 1, 3 one by one in the unit’s place to find three two-digit numbers.
Thus, we get the two-digit numbers 20, 21, and 23.
Case 3: The digit at tens place is 3.
Now, we have used the digit 3 at the tens place.
The remaining digits are 0, 1, and 2.We will put the remaining digits 0, 1, 2 one by one in the unit’s place to find three two-digit numbers.
Thus, we get the two-digit numbers 30, 31, and 32.Therefore, from the three cases, we have the numbers 10, 12, 13, 20, 21, 23, 30, 31, and 32.

This is the sample space $S$. We can write the sample space as
S = \left\\{ {10, 12, 13, 20, 21, 23, 30, 31, 32} \right\\}
The number of elements in the sample space is 9.
Hence, we get $n\left( S \right) = 9$.

Note:
Sample space is a set of all possible outcomes of a random experiment. Here, the first thing we need to keep in mind is that a two-digit number can never start with 0. Another common mistake we can commit is to include the numbers 11, 22, and 33, thus ignoring the given condition ‘without repetition’. These should not be added in the answer unless the question says ‘with repetition'.