Exercise 874Combinatorics › Counting principlesHigh schoolAdded 3 Nov 2026

Ordering books on a shelf

#permutations#block arrangement#product

Statement

There are $3$ math books, $2$ physics books, and $2$ chemistry books to be arranged in a row, but all books of the same subject must stay together. How many arrangements are possible?

Hint

Treat each group as a block. Arrange the blocks, then the books within each block.

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Solution

Blocks. $3$ blocks (math, physics, chemistry).

Arrange blocks. $3!=6$ ways.

Within blocks. Math: $3!=6$ ; physics: $2!=2$ ; chemistry: $2!=2$ .

Total. $6\cdot6\cdot2\cdot2=144$ .

Conclusion. There are $144$ arrangements.

Ordering books on a shelf

Statement

Solution

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