<p>The number of selections of 4 letters taken from the word COLLEGE is equal to</p>
<p>(A) 24</p>
<p>(B) 18</p>
<p>(C) Coefficient of \(x^4\) is \((1+x)(1+x+x^2)\)</p>
<p>(D) Coefficient of \(x^4\) in \((1+x)^3(1+x+x^2)\)</p>
Step-by-Step Solution
Key Concept: Use generating functions where exponents represent letter counts and coefficient of \(x^4\) gives total selections.
<p>The word COLLEGE has letters: C, O, L, L, E, G, E. We need to count selections of 4 letters considering multiplicities: C(1), O(1), L(2), E(2), G(1).</p><p>Using generating functions, the coefficient of \(x^4\) in \((1+x)(1+x+x^2)(1+x+x^2)(1+x)(1+x)\) gives all possible selections.</p><p>This simplifies to finding coefficient of \(x^4\) in \((1+x)^3(1+x+x^2)^2\), which equals 18.</p>
Correct Answer: B