Permutations & Combinations
Binomial Coefficients and Their Interpretations
Grade 11
Question:
<p>The coefficient of \(x^{50}\) in the expansion of \(\sum_{k=0}^{100} \binom{100}{k}(x-2)^{100-k}3^k\) is also equal to:</p>
<p>(a) Number of ways in which 50 identical books can be distributed in 100 students, if each student can get at most one book.</p>
<p>(b) Number of ways in which 100 different white balls and 50 identical red balls can be arranged in a circle, if no two red balls are together.</p>
<p>(c) Number of dissimilar terms in \((x_1 + x_2 + x_3 + \ldots + x_{50})^{51}\).</p>
<p>(d) \(\frac{2 \times 6 \times 10 \times 14 \times \ldots \times 198}{50!}\)</p>
Step-by-Step Solution
Key Concept: Recognize the expansion as a binomial and find the coefficient; then match it with combinatorial interpretations.
<p>Using the binomial expansion $\sum_{k=0}^{100} \binom{100}{k}(x-2)^{100-k}3^k = (x-2+3)^{100} = (x+1)^{100}$.</p><p>The coefficient of $x^{50}$ in $(x+1)^{100}$ is $\binom{100}{50}$.</p><p>This equals: (a) the number of ways to choose 50 students from 100 to each receive one book (selecting 50 from 100), and (d) the given product formula which simplifies to $\binom{100}{50}$.</p>
Correct Answer: a, d