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<blockquote data-quote="tarchon" data-source="post: 1403525" data-attributes="member: 5990">I am a math guy, and I have to say there are few things in the world more boring than looking for somebody else's arithmetic error. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile&nbsp; &nbsp; :)"&nbsp; data-smilie="1"data-shortname=":)" />The problem with combinatorics methodologies is that they usually only give neat expressions in very limited circumstances.&nbsp;&nbsp; The way a pro does this kind of thing is to first glance at the problem, then decide if there's likely to be a neat exact solution in terms of commonly used expressions, which there probably isn't (this decision is largely based on experience and intuition).&nbsp; If there isn't, the pro either does it numerically on a computer (for finite and small event spaces like this, it's easy to solve it exactly) or makes approximations that will give a simple solution good enough for what's needed.&nbsp; This last approach is the real art form.The computer route takes like 5 seconds to run and ten minutes to write.</blockquote>

[QUOTE="tarchon, post: 1403525, member: 5990"] I am a math guy, and I have to say there are few things in the world more boring than looking for somebody else's arithmetic error. :) The problem with combinatorics methodologies is that they usually only give neat expressions in very limited circumstances. The way a pro does this kind of thing is to first glance at the problem, then decide if there's likely to be a neat exact solution in terms of commonly used expressions, which there probably isn't (this decision is largely based on experience and intuition). If there isn't, the pro either does it numerically on a computer (for finite and small event spaces like this, it's easy to solve it exactly) or makes approximations that will give a simple solution good enough for what's needed. This last approach is the real art form. The computer route takes like 5 seconds to run and ten minutes to write. [/QUOTE]

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