JMC 2026 Q25

追加: 2026-05-07

[00:00:10]So this question concerns four runners.

[00:00:13]They have a race and they arrive in some order and the question allows two of them to be tied.

[00:00:19]You've got to determine how many possible orderings there are for these runners.

[00:00:25]Well, let's do the easy thing first.

[00:00:26]Let's first imagine that there are no ties.

[00:00:30]Who comes first?

[00:00:32]Well, it could be any one of the four of them. So there are four choices what comes first.

[00:00:37]And that person is now out of the pool of people who might come second. There are three choices of who comes second.

[00:00:45]>> [snorts] >> And then there are two more choices for who comes third.

[00:00:53]And you can either stop there or you can say there's only one choice of who comes last. There's only one person left.

[00:01:00]Now, there's some notation for this kind of falling product. It's four factorial.

[00:01:08]You may not have come across it yet, but you you will.

[00:01:13]This number when you calculate it is 24.

[00:01:17]So there are 24 ways of the race finishing if there are no ties.

[00:01:23]Now, let's let's consider any particular running order or finishing order. A comes first, B comes second, C comes third, D comes fourth.

[00:01:35]If if we allow ties, then maybe A and B tie or B and C tie or C and D tie.

[00:01:43]So you you might first think that the number of the number of possibilities for a race involving a tie would be well, there are 24 initial orders.

[00:01:56]We've already worked out where they all finish differently, but then when you start pushing two of them together, you're increasing the number of options by three.

[00:02:07]>> [snorts] >> Except that's not quite right.

[00:02:10]Because for example, if you started with B finishing before A finishing before C and finishing before D, then pushing those two together so that they formed a tie would give the same result as pushing those two together. And that would apply no matter which finishing order there was and it no matter which two you push together. So you're actually counting everything twice.

[00:02:36]So we better divide by two.

[00:02:41]And that's the correct number of races which race finishes which involve a tie.

[00:02:48]Now when you work that out, it's 36.

[00:02:53]The total number of options is the number of races race finishes which don't involve a tie plus the number which do involve a tie. You add those numbers up.

[00:03:05]You get 60.

[00:03:07]And that's option B.