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How many ways can I arrange, say, 3 letters and 3 digits, if the letters must appear in groups?

For starters, I know we can do $26^3 * 10^3$ to represent a 6 letter string where letters together are followed by 3 digits. Then, by inspection, I can see that there are 4 places the chunk can go, so my final answer (I think) would be $26^3 * 10^3 * 4$. Is there a general way to do these problem without examining each on a case by case basis?

Thanks!

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    I think your approach to the "Three letters and three digits, and the letters have to be together" case is exemplary. – 2017-01-14

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Let's say that you have $ x $ letters and $ y $ numbers. If you want to create a string where all letters appear in one group, there are a total of $ 26^x \times 10^y \times (y+1) $ ways to create such an arrangement. Note that the $ y + 1 $ term comes from the fact that you can place your group of letters before the $ 1^{st} $ number, before the $ 2^{nd} $ number, ... , before the $ y^{th} $ number, and after the $ y^{th} $ number.

In your case, you have $ x = 3 $ and $ y = 3 $, so you get: $ 26^x \times 10^y \times (y+1) = 26^3 \times 10^3 \times 4 $.

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    When you say before the 1st letter, 2nd letter, etc., did you mean digit? Other than that, it makes complete sense why the number of positions you can place your group of letters in is constrained by the number of digits you have. Thanks! – 2017-01-16
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    @tomatoBisque Yes, I meant after the 1st number, 2nd number, etc. I've made the edit. Thanks for pointing that out! – 2017-01-16