How many different five-digit number license plates can be made if (i) the first digit cannot be zero, and the repetition of digits is not allowed, (ii) the first-digit cannot be zero, but the repetition of digits is allowed?

Solution:

(i) We know that the initial digit of the license plates cannot be zero. It is also forbidden to repeat digits.

Let’s say there are five boxes. The first box can be filled with any of the nine digits, thus ⁹C₁ is an option.

Similarly, the second box can be filled with any of the nine digits available, so ⁹C₁ is an option.

The third box can be filled with any of the eight digits available, therefore ⁸C₁ is a possibility.

The fourth box can be filled with any of the seven digits available, therefore ⁷C₁ is a possibility.

The fifth box can be filled with any of the six digits available, therefore ⁶C₁ is an option.

As a result, the total number of outcomes is ⁹C₁ × ⁹C₁ × ⁸C₁ × ⁷C₁ × ⁶C₁ = 9 × 9 × 8 × 7 × 6 = 27,216

(ii) We know that zero cannot be the first digit of the license plates. And the repetition of digits is allowed.

Let us assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is ⁹C₁

Similarly, the second box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

The third box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

The fourth box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

The fifth box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

Hence, the total number of possible outcomes is  ⁹C₁ × ¹⁰C₁ × ¹⁰C₁ × ¹⁰C₁ × ¹⁰C₁ = 9 × 10 × 10 × 10 × 10 = 90,000