There are three phrases that people use on quiz shows frequently. They all prompt me to shout, or at least mutter, at the TV.
The first is, “I wouldn’t have got that in a million years”. I always point out that a million years is a really long time. If you uttered every combination of vowel and consonant sounds, up to 30 syllables, for a million years you would get the answer to any quiz question. You might need an infinite number of monkeys sat at an infinite number of typewriters for an infinite amount of time to come up with the Complete Works of Shakespeare. But a million years will be enough to come up with the answer to any question.
I am especially inclined to shout at the TV when someone responds in this way to a question that only has a limited number of possible answers. There are four possible answers to each question on “Who wants to be a millionaire?”, and three in the “Chaser” rounds on “The Chase”. A million years to cycle through three or four answers? You could spend a thousand years on each possible answer, and even if you end up with the fourth answer on “Who wants to be a millionaire?” you’ll still have 996,000 years left. Honestly, you would get it in a million years.
Similarly when someone says, “You’re not a million miles away” I am always prompted to say, “Nothing on earth is a million miles away”. If you travel from earth towards the moon you will get there after 239,000 miles or so (the average distance between here and there) and still have more than 760,000 miles to go.
I realize that “Not a million miles away” is a figurative expression, but sometimes it’s used literally, on Richard Osman’s “House of Games” for example. In the round called “Where is Kazakhstan?” contestants are shown a map (the USA, or Europe perhaps) and have to place a mark on a specific point. It could be “Yale University” or “The major city that completes the title in the 1944 Judy Garland musical ‘Meet me in….’”. The answer might be met with, “You’re not a million miles away”, in which case I will be saying, out loud, “Nowhere on earth is a million miles away”. No matter where the contestants put their X they will be within a million miles. If it’s a map of the UK they will be within a thousand miles.
The third, and most troubling, phrase that I hear frequently on quiz shows relates to the likelihood of someone being killed if they don’t get an answer right. A typical example would be a contestant being asked a question Arsenal football club and saying, “Oh my dad’s a massive Arsenal fan; he’ll kill me if I get this wrong”. If my son is in the room with me I’ll turn to him and say, “Someone’s going to get murdered for not knowing the answer to a quiz question. Imagine that.” It’s just a TV quiz. Nobody should be murdered for failing to get a quiz question right. If there is a chance of it happening they should get away as quickly as possible, maybe as far away as possible. A million miles away.
The Compartments 
Had some fun testing an LLM with this, quite interesting …That is a fantastic setup for a rebuttal that uses combinatorial mathematics to argue against the casual use of large numbers like “a million years” in language!
The core of the rebuttal will be demonstrating that the number of possible 30-syllable utterances is vastly larger than the number of seconds in a million years, and that Shakespeare’s works are exponentially more complex than the quiz answer. Here is the construction of your rebuttal based on the steps you outlined.
The Combinatorics of Language and Literature
The claim that “a million years will be enough to come up with the answer to any question” by uttering combinations of sounds is incorrect because it drastically underestimates the sheer number of possible combinations available in a 30 syllable utterance. Similarly, the “infinite monkeys” analogy is often misunderstood; achieving the Complete Works of Shakespeare is far more attainable than the claim suggests, though still requiring vast resources.
English language has approximately 24 distinct consonant phonemes, and 15 distinct vowel phonemes, including monophthongs and the major diphthongs. Triphthongs are usually considered sequences of a monophthong and a diphthong, so we use the lower number.
A syllable in English generally takes one of the following forms: {V}(Vowel), {CV}(Consonant-Vowel), {VC}(Vowel-Consonant), or {CVC}(Consonant-Vowel-Consonant)
Your sum aims to calculate the total number of utterances using N_V = 15 vowel sounds and N_C = 24 consonant sounds across a 30 sound sequence, assuming that any combination of V and C is a “feasible answer.”
Conclusion on Quiz Answers: The number of possible 30 sound utterances is approximately 1.93 x 10^34 times larger than the number of seconds in a million years.
The OP suggests the Infinite Monkey Theorem applies to Shakespeare. We can make this concrete.
Character Space and Work SizeCharacter Set: 26{a-z} + 26{A-Z} + 1{space} = 53
Characters in Shakespeare’s Complete Works: An accepted estimate for the number of characters in the First Folio (the collected plays) is approximately M ~ 5 million characters.
To guarantee that the chimp types the sequence correctly on the first attempt, we need to try 53 ^{M = 5E6} times.
However, the problem specifies a scenario where the number of chimps equals the typing time in seconds.
The time required is 1E4,300,000 centuries,and this requires 1E4,310,000 chimps typing once a second.
While the task is technically finite, the magnitude is so large that it is effectively impossible, meaning the “infinite monkeys” analogy correct. The claim that “a million years” is sufficient for the quiz answer is vastly more false in magnitude than the perceived impossibility of the Shakespeare task.
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