# How many ways are there to make a schedule of 6 subjects

There are 6 different subjects including maths and physics.

How many ways are there to make a schedule of 6 subjects such that physics follows maths?

Actually I tried to denote maths as M and physics as P. Then I started to count like this.

MXXXXX - number of ways - 5!

XMXXXX - number of ways - 4*1*4*3*2*1 = 96

XXMXXX - number of ways - 4*3*1*3*2*1 = 72

XXXMXX - number of ways - 4*3*2*1*2*1 = 48

XXXXMX - number of ways - 4*3*2*1*1*1 = 24

And then sum them up, getting 360, but actually the correct answer to the problem is 240.

Can anyone help me solve this problem?

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Follow immediately after, or somewhere after? If immediately after, then simply "attach them together" as a single subject, then you have $5$ subjects instead of $6$, then there are $5!$ ways to arrange them. If somewhere after, then simply divide the total number of ways ($6!$) by $2$, since physics can equally likely come before math and after math. – barak manos Mar 20 at 12:34
Are you sure it isnt $120$ – Archis Welankar Mar 20 at 12:34
Yes correctly asked @barak manos – Archis Welankar Mar 20 at 12:35
@barakmanos You should turn your symmetry argument into an answer. – N. F. Taussig Mar 20 at 12:39
Maybe the intended wording was "That one follows the other" or something so that P follows M directly or M follows P directly, in that case it would be $5! *2=240$ – Ivo Beckers Mar 20 at 17:46

The total number of ways for arranging those $6$ subjects is $6!=720$.

Physics may equally likely appear before Math and after Math.

So there are $720/2=360$ arrangements with Physics after Math.

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There are six slots to fill. We can fill two of those six slots with mathematics and physics in $\binom{6}{2}$ ways. Since physics must follow maths, there is only one way to arrange mathematics and physics in those slots. For each of these ways of scheduling maths and physics, there are $4!$ ways of arranging the other subjects in the four remaining slots. Hence, the number of possible schedules in which physics follows mathematics is $$\binom{6}{2} \cdot 4! = 360$$ as you found.

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$$\binom62\times4!=15\times24=360$$

From $6$ spots first select $2$ for maths and physics. Then place the other subjects in some order. There are $4!$ orders.

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Actually I understand that my approach to the problem is correct. However the issue is that the correct answer is 240 rather than 360. Could you please explain why is the answer 240. – Marie Mar 20 at 12:50
@Marie Sometimes the solution key is incorrect. – N. F. Taussig Mar 20 at 12:51
Everybody here agrees with your answer. I don't think that's a coincidence ;) and fully agree with @N.F.Taussig. No idea how "they" come to $240$. – drhab Mar 20 at 12:54

There are 6! = 720 ways to order the six subjects. Half these orderings will have math before physics.

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Choose 2 slots for m and p: $\binom{6}{2}$, this accounts for the fact that p follos m. For each such choice you have 4! allocations of other subjects

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The correct answer is 240 because:

Imagine the math and physics classes together as a "package", they can be thought of as one two hour long class following each other. Considering that, there will be 5!=120 different schedules. However, within the "physics-math class", you can take physics first or math first, so there are 2!=2 ways to do that. Therefore, the total possible schedules are:

       5!2!=240 schedules possible


We solved a very similar problem in precalculus recently

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You overlooked the requirement that physics follows maths. – N. F. Taussig Mar 21 at 1:52
No, read it over again. That is where the 2! comes in. – Suhas Mar 21 at 1:53
I was not referring to your answer. The question states "How many ways are there to make a schedule of 6 subjects such that physics follows maths?" – N. F. Taussig Mar 21 at 1:55
So did you get the answer to be 5!=120 schedules – Suhas Mar 21 at 1:56
That would be the correct answer if physics had to immediately follow maths. If physics has to appear somewhere after maths in the schedule, then the answer is $360$. The wording is, alas, ambiguous. – N. F. Taussig Mar 21 at 1:59