Permutations

[thestag]

How many ways can 21 English & 19 Spanish books be arranged in a way that no two Spanish books are together?

 

[Deleted member 4993]

How many ways can 21 English & 19 Spanish books be arranged in a way that no two Spanish books are together?

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this assignment.

 

[thestag]

I have tried to solve by finding P(19,22) But I don't think that's it. I mean I have no clue

 

[Steven G]

Think about how you would arrange the books. Think about how you would arrange the books if you were to violate the given restraint.

 

[pka]

How many ways can 21 English & 19 Spanish books be arranged in a way that no two Spanish books are together?

This is a classic separator problem.. In some material on combinatorics it is know as the stars & bars problem.
Consider $****||||||$, four stars and six bars. that string can be arranged in $\dfrac{10!}{6!\cdot 4!}=210$ ways. Here is one way: $*|~|~|**|~|*|$ We can see that the six bars create seven places $\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_$ in which to place a star.
Thus there are $\mathcal{C}_4^7=35$ ways to separate the stars.
If we have $N~*'s$ and $k~|'s$ then if $k\ge N-1$ we can separate the stars using the bars in $\mathcal{C}_N^{k+1}$ ways.

 

[thestag]

This is a classic separator problem.. In some material on combinatorics it is know as the stars & bars problem.
Consider $****||||||$, four stars and six bars. that string can be arranged in $\dfrac{10!}{6!\cdot 4!}=210$ ways. Here is one way: $*|~|~|**|~|*|$ We can see that the six bars create seven places $\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_|\_\_\_\_$ in which to place a star.
Thus there are $\mathcal{C}_4^7=35$ ways to separate the stars.
If we have $N~*'s$ and $k~|'s$ then if $k\ge N-1$ we can separate the stars using the bars in $\mathcal{C}_N^{k+1}$ ways.

Thanks for your explanation. This is a whole new concept that I didn't know of. I think yt might be a good place to start.