I found the probability topic in stats and mechanics to be very interesting, and I attempted to try using it to calculate the odds of winning the UK National Lottery, but failed. My calculation was (1/59*1/58*1/57*1/56*1/55*1/54). The reason was that in the NL there are 6 balls dropped, and the possible outcomes range from 1 to 59. I reasoned that since all 6 need to match, I could assume that each outcome could be treated as an isolated selection, and that my first ball match odds were 1/59 as a result, then if the first ball matches (which it must), the next is 1/58 and so on. The odds I calculated were orders of magnitude less likely than the correct value. What is wrong with my attempt, and why is the correct formula quoted as "59!/(6!*(59-6)!)"?
2026-02-23 04:50:18.1771822218
UK Lottery Odds Calculation Error
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Firstly, note the answer you obtained is $\frac{53!}{59!}$, while the probability you should have obtained is $\frac{53!6!}{59!}$ (you've stated this fraction upside down, where it gives the number of possible outcomes). The reason you're wrong by a factor of $6!$ is because the order in which the numbers fall is irrelevant. This is the combinations-permutations distinction; in technical terms, $\frac{59!}{53!6!}={}^{59}C_6$ while $\frac{59!}{53!}={}^{59}P_6$.