A letter lock has $3$ rings containing $6$ different letters. No $2$ rings have the same letters. How many different combinations of passwords is possible?

  1. $15600$

  2. $17576$

  3. $\binom{26}{6}\binom{20}{6}\binom{14}{6}6^3$

  4. $\binom{26}{6}^36^3$

I am getting $3$ as the answer but the answer given is $15600$... please explain what's the correct one?

  • $\begingroup$ The question is not at all clearly formulated. Not clear what "combinations of passwords" are, not whether there is any relation between passwords and rings, and if so which relation. Also the title is not very helpful. $\endgroup$ – Marc van Leeuwen Jun 3 '16 at 14:58

There are three independent rings, each with 6 options. So $6^3$ is the right answer (the order matters). This assumes the rings are fixed and known in advance.

But this is not among the options.

Your answer (nr 3) is the number of ways we can pick the letters for each ring first (so start with blank rings), and then times $6^3$ for each combination.

But this overcounts. The combination ABC could occur in many different ways.

The net effect of picking rings and letters for each is that every combination of three different letters could be a combination for lots of ring configurations so the answer would then $26 \cdot 25 \cdot 24$.

So if the rings are fixed, the answer is $6^3$, if they're configurable, the answer is $26 \cdot 25 \cdot 24$. The question is unclear, IMHO, and not very good.

  • $\begingroup$ can you please elaborate "ABC will occur in many different ways" statement? $\endgroup$ – ghostrider Jun 3 '16 at 14:35
  • 1
    $\begingroup$ @ghostrider if you pick rings AIJDEF BYKLMN CRSTUV you have combination ABC, but also if you chose the rings AGJDEF BZKLMN CRSTUV (which is different in the choices of rings), etc. You can see that ABC can occur from many choices of rings. $\endgroup$ – Henno Brandsma Jun 3 '16 at 14:40

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