Question 6

Explanation

To form a distinct four-digit odd number from the digits $1, 2, 3, 4, 5, 6$, we can use the filling slots method.

There are $4$ digit positions: Thousands, Hundreds, Tens, and Units.

Step 1: Filling the Units Position

For the number to be odd, the units digit must be an odd number. From the available digits ($1, 2, 3, 4, 5, 6$), the odd numbers are $1, 3, 5$.

So, there are $3$ choices for the units position.

Step 2: Filling the Thousands Position

After one digit is used for the units position, there are $6 - 1 = 5$ digits remaining for the thousands position (since the digits must be distinct).

So, there are $5$ choices for the thousands position.

Step 3: Filling the Hundreds Position

After two digits are used (for units and thousands), there are $6 - 2 = 4$ digits remaining.

So, there are $4$ choices for the hundreds position.

Step 4: Filling the Tens Position

After three digits are used, there are $6 - 3 = 3$ digits remaining.

So, there are $3$ choices for the tens position.

Total Calculation

The number of numbers that can be formed is the product of the number of choices for each position:

So, there are $180$ numbers that can be formed.