Binomial Theorem
Binomial Coefficient Properties
Grade 11
Question:
<p>Number of values of <i>r</i> satisfying the equation <span class="math">\binom{69}{3r-1} + \binom{69}{r} = 2\binom{69}{3r} + \binom{69}{r^2+1}</span> is</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 3</p>
<p>(d) 7</p>
Step-by-Step Solution
Key Concept: Apply symmetry property of binomial coefficients and validity constraints to find all integer solutions.
<p><strong>Solution:</strong> Use the property that <span class="math">\binom{n}{r} = \binom{n}{n-r}</span> and the constraint that all binomial coefficient arguments must be valid (between 0 and 69). Find values of <i>r</i> satisfying these conditions.</p>
Correct Answer: C