Permutations & Combinations
Grade None
Question:
<p>Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to</p>
<p style="display:inline">18</p>
<p style="display:inline">12</p>
<p style="display:inline">16</p>
<p style="display:inline">15</p>
Step-by-Step Solution
Key Concept: The problem requires finding the number of ways to partition the integer 8 into at most 4 parts because both the books and the shelves are identical.
<p>8 identical objects into 4 identical shelves where any number of shelves may remain empty is given by<br />
<span class="math-tex">\(008,0017,0026,0035,0044=5\)</span><br />
<span class="math-tex">\(\left.\begin{array}{llll}0 & 1 & 1 & 6 \\ 0 & 1 & 2 & 5 \\ 0 & 1 & 3 & 4 \\ 0 & 2 & 2 & 4\end{array}\right]\)</span><br />
0224<br />
<span class="math-tex">\(\left.\begin{array}{llll}1 & 1 & 1 & 5 \\ 1 & 1 & 2 & 4 \\ 1 & 1 & 3 & 3 \\ 1 & 2 & 2 & 3\end{array}\right]\)</span><br />
<u>2222-1</u><br />
Total <span class="math-tex">\(=15\)</span></p>
Correct Answer: D