A product in ordinary arithmetic is just a repeated sum: 3×5=3+3+3+3+33×5=3+3+3+3+3. The same goes for the product rule in combinatorics: it is just a repeated application of the sum rule.

I'm going to assume you're happy with the sum rule: this is the idea that if we're counting things (arrangements, outcomes, whatever) that fall under several non-overlapping cases, we count each case separately, then add them up.

The product rule happens when there are many non-overlapping cases that are counted in the exact same way. In that case, you're going to add them all up, and if there are nn cases and kk outcomes in each case, you're going to get k+k+⋯+kn=nkk+k+⋯+k⏟n=nk.

For example, suppose we want to count the number of permutations of the letters A,B,C,D. We split these up into 44 cases: the case where A is first, the case where B is first, the case where C is first, and the case where D is first. Then we notice that each of these cases is identical: in each of them, we have three letters left over, and three slots to place them in. This is a problem you've already solved: there are 3⋅2⋅1=63⋅2⋅1=6 permutations of A,B,C, so there are 66 permutations of any three letters. If we have 44cases and 66 permutations in each case, then there are 4⋅64⋅6 permutations in total.

A large part of the product rule is just becoming familiar enough with this idea that you do it on autopilot. If you want to know the number of 44-letter sequences from the English alphabet, the argument above suggests that you

1. Split them up into 2626 cases based on the first letter.
2. Realize that each case is identical: counting the number of 33-letter sequences.
3. Split up each case into d="MathJax-Element-17-Frame" class="MathJax" style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; box-sizing: inherit; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float