Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

Permutations and Combinations PYQ — JEE Main 2024 · 29 Jan · Shift 2 · Q31 with Solution

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Correct answer: D

Interpret the problem

We have:

$8$ identical books
$4$ identical shelves
shelves may be empty

Since both the books and shelves are identical, an arrangement is determined only by how $8$ is split into at most 4 parts, where parts represent the numbers of books on the nonempty shelves.

So we need the number of partitions of $8$ into at most $4$ nonnegative groups, ignoring order.

Convert to partition counting

Let the numbers of books on the shelves be $a_1,a_2,a_3,a_4$ with

\qquad a_i\ge 0$$ But since shelves are **identical**, arrangements differing only by permutation are the same. Hence we count unordered partitions of $8$ into at most $4$ parts. --- 3. **List all partitions of $8$ into at most 4 parts** ### Partitions into 1 part: - $8$ Count = $1$ ### Partitions into 2 parts: - $7+1$ - $6+2$ - $5+3$ - $4+4$ Count = $4$ ### Partitions into 3 parts: - $6+1+1$ - $5+2+1$ - $4+3+1$ - $4+2+2$ - $3+3+2$ Count = $5$ ### Partitions into 4 parts: - $5+1+1+1$ - $4+2+1+1$ - $3+3+1+1$ - $3+2+2+1$ - $2+2+2+2$ Count = $5$ --- 4. **Total number of arrangements** $$1+4+5+5=15$$ So the number of ways is $$\boxed{15}$$ --- 5. **Check options** - A: $18$ ❌ - B: $16$ ❌ - C: $12$ ❌ - D: $15$ ✅ Therefore, the correct option is $$\boxed{\text{D}}$$

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View written solutionFree

Correct answer: D

Interpret the problem

We have:

$8$ identical books
$4$ identical shelves
shelves may be empty

Since both the books and shelves are identical, an arrangement is determined only by how $8$ is split into at most 4 parts, where parts represent the numbers of books on the nonempty shelves.

So we need the number of partitions of $8$ into at most $4$ nonnegative groups, ignoring order.

Convert to partition counting

Let the numbers of books on the shelves be $a_1,a_2,a_3,a_4$ with

\qquad a_i\ge 0$$ But since shelves are **identical**, arrangements differing only by permutation are the same. Hence we count unordered partitions of $8$ into at most $4$ parts. --- 3. **List all partitions of $8$ into at most 4 parts** ### Partitions into 1 part: - $8$ Count = $1$ ### Partitions into 2 parts: - $7+1$ - $6+2$ - $5+3$ - $4+4$ Count = $4$ ### Partitions into 3 parts: - $6+1+1$ - $5+2+1$ - $4+3+1$ - $4+2+2$ - $3+3+2$ Count = $5$ ### Partitions into 4 parts: - $5+1+1+1$ - $4+2+1+1$ - $3+3+1+1$ - $3+2+2+1$ - $2+2+2+2$ Count = $5$ --- 4. **Total number of arrangements** $$1+4+5+5=15$$ So the number of ways is $$\boxed{15}$$ --- 5. **Check options** - A: $18$ ❌ - B: $16$ ❌ - C: $12$ ❌ - D: $15$ ✅ Therefore, the correct option is $$\boxed{\text{D}}$$

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