Question

A car number plate consists of a letter, followed by five numbers, and ended with three letters. How many car number plates can be formed? if required there must not be the same letter and no equal lift, and how many car number plates can be formed ?

Answer 1

Example: a12345bcd

Note: there are 26 letters and 10 digits available

1) If numbers and letters can be repeated

We have 4 positions for letters and 5 positions for numbers.

So, we can choose 1 out of 26 letters to first position, 1 out of 10 numbers for each of next 5 positions, and 1 out of 26 letters for each of last 3 positions.

In this way we have: $26\cdot10\cdot10\cdot10\cdot10\cdot10\cdot26\cdot26\cdot26=26^4\cdot10^5 =45,697,600,000$ variants.

2) If no repeating is allowed.

In this case amount of numbers and letters we can choose will decrease by 1 on each step. (If we already have 'a' in the plate, we can't put it again)

$26\cdot10\cdot9\cdot8\cdot7\cdot6\cdot25\cdot24\cdot23=10,850,112,000$

Other solution. We can say, that we choose 4 different letters and 5 different numbers. And than we arrange them on their places.

$\begin{pmatrix} 26\\4 \end{pmatrix}\cdot4!\cdot \begin{pmatrix} 10\\5 \end{pmatrix}\cdot5!$

Actually, I would not recoment this solution. It looks like round trip. But may be more convenient in some other tasks.