<p>Find the value of <i>r</i>, if <span style="math">\(^8P_5 + 5 \cdot \,^8P_4 = \,^9P_r\)</span></p>
Step-by-Step Solution
Key Concept: Factor out common permutation terms and use the relationship between permutation formulas to solve for r.
<p><strong>Solution:</strong></p><p>Given: <span style="math">$^8P_5 + 5 \cdot \,^8P_4 = \,^9P_r$</span></p><p>Dividing by <span style="math">$^8P_4$</span>: <span style="math">$\frac{^8P_5}{^8P_4} + 5 = \frac{^9P_r}{^8P_4}$</span></p><p>Simplifying: <span style="math">$^8P_4(8 - 5 + 1 + 5) = \,^9P_r$</span></p><p><span style="math">$9 \cdot \,^8P_4 = \,^9P_r$</span></p><p><span style="math">$^9P_5 = \,^9P_r$</span></p><p>Therefore, <span style="math">$r = 5$</span></p>
Correct Answer: 5