In how many ways can $4$ Latin, and $1$ English book be placed on a shelf so that the English book is always in the middle, the selection made from $7$ Latin and $3$ English books?

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Solution

We want an arrangement of books as below

Latin Latin English Latin Latin
The middle book can be chosen in 3 ways as we want an english books

The remaining 4 books can be chosen and arranged from 7 Latin books in $^{7} P_{4} = 840$ ways

Hence 5 books can be arranged in $840 \times 3 = 2520$ ways

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In how many ways can 4 Latin, and 1 English book be placed on a shelf so that the English book is always in the middle, the selection made from 7 Latin and 3 English books?

We want an arrangement of books as below Latin Latin English Latin LatinThe middle book can be chosen in 3 ways as we want an english booksThe remaining 4 books can be chosen and arranged from 7 Latin books in 7P4=840 waysHence 5 books can be arranged in 840×3=2520 ways

In how many ways can $4$ Latin, and $1$ English book be placed on a shelf so that the English book is always in the middle, the selection made from $7$ Latin and $3$ English books?

We want an arrangement of books as below

Latin Latin English Latin Latin
The middle book can be chosen in 3 ways as we want an english books

The remaining 4 books can be chosen and arranged from 7 Latin books in $^{7} P_{4} = 840$ ways

Hence 5 books can be arranged in $840 \times 3 = 2520$ ways