Binomial Theorem
Binomial Coefficient Identities
Grade 11
Question:
<p>If <span class="math">f(n) = \sum_{i=0}^{30}\binom{30}{i}</span>, then</p>
<p>(a) maximum value of <i>f</i>(<i>n</i>) is <span class="math">\binom{50}{25}</span></p>
<p>(b) <i>f</i>(0) + <i>f</i>(1) + <i>f</i>(2) + ... + <i>f</i>(50) = 2<sup>50</sup></p>
<p>(c) <i>f</i>(<i>n</i>) is always divisible by 50</p>
<p>(d) <i>f</i><sup>2</sup>(0) + <i>f</i><sup>2</sup>(1) + <i>f</i><sup>2</sup>(2) + ... + <i>f</i><sup>2</sup>(50) = <span class="math">\binom{100}{50}</span></p>
Step-by-Step Solution
Key Concept: Apply Vandermonde's convolution identity to evaluate the sum of squares of binomial coefficients.
<p><strong>Solution:</strong> Use Vandermonde's identity: <span class="math">\sum_{k=0}^{n}\binom{n}{k}^2 = \binom{2n}{n}</span>. This gives <span class="math">\sum_{i=0}^{50}f^2(i) = \binom{100}{50}</span>.</p>
Correct Answer: D