Permutations & Combinations
Combinations with Restrictions
Grade 11
Question:
<p>A library has 5 indistinguishable physics books, 4 indistinguishable mathematics books and 3 indistinguishable chemistry books. In how many distinguishable ways can a student take home 6 books?</p>
<p>(A) 16</p>
<p>(B) 18</p>
<p>(C) 19</p>
<p>(D) 20</p>
Step-by-Step Solution
Key Concept: Convert the problem to finding non-negative integer solutions with upper bounds; use generating functions to count valid distributions.
<p>Let \(p\), \(m\), \(c\) denote the number of physics, mathematics, and chemistry books taken respectively.</p><p>We need: \(p + m + c = 6\) where \(0 \leq p \leq 5\), \(0 \leq m \leq 4\), \(0 \leq c \leq 3\).</p><p>Using generating functions or systematic enumeration:</p><p>The coefficient of \(x^6\) in \((1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4)(1+x+x^2+x^3)\) gives 19 ways.</p>
Correct Answer: C