Permutations & Combinations
Grade 11
Question:
<p>The number of ways in which n distinct objects are placed in two distinguishable boxes so that no box remains empty, is</p>
<p style="display:inline">2<sup>n</sup> - 2</p>
<p style="display:inline">n<sup>2</sup></p>
<p style="display:inline">n!</p>
<p style="display:inline">2<sup>n</sup></p>
Step-by-Step Solution
Key Concept: Calculate the total number of distributions by giving each object two choices and subtract the two specific scenarios where all objects are in the same box.
<p>Let B<sub>1</sub> and B<sub>2</sub> be two boxes<br />
Each object has two options (B<sub>1</sub> or B<sub>2</sub>) to go.<br />
<span class="math-tex">$\Rightarrow$</span> There are <span class="math-tex">$\underbrace{2 \times 2 \times \ldots \times 2}_{n \text { times }}$</span> = 2<sup>n</sup> ways to place n objects<br />
These 2<sup>n</sup> cases include two cases in which either of the boxes remains empty.<br />
<span class="math-tex">$\Rightarrow$</span> Required number = 2<sup>n</sup> - 2</p>
Correct Answer: A