Let's think about it this way.
1) Pick a letter, any letter, and put it in the first position. How many choices do you have? 26.
2) Pick any letter, even the same letter as the last step, and put it in the second position. How many choices? 26.
Now pause for a minute. How many different two-letter possibilities are there in all? If you pick an "A" for the first, there are 26 possibilities that start with "A". If you pick a "B", then there are 26 other possibilities that start with "B". How many times are there 26 possibilities? One for each of "A" to "Z" - that's 26 times. And what is 26 times 26? It is 676. But the important principle here is that you multiply each position's possibilities together.
So let's go on:
3) Pick another letter. 26 choices. So the total number of possibilities is 26 times the previous number 676. Let's right this as 26*262 = 263. That's it for the letters, now for the numbers.
4) How many choices do you have for the fourth position (the first position with numbers)? The digits 0-9, so that's ten possibilities. Multiply that by the previous total and you get 263 * 10.
5-7) Do this one more time for each of the other three positions and you multiply by 10 three more times. Now your total possibilities is 263 * 104.
Hope this helps!