<p>If the sum of binomial coefficients in the expansion of \((x + y)^n\) is 1024 then the greatest binomial coefficient occurs in the ______ th term.</p>
Step-by-Step Solution
Key Concept: The sum of all binomial coefficients in (x+y)^n equals 2^n (found by setting x=y=1). Use this to find n, then determine which term has the maximum binomial coefficient.
<p><strong>Step 1:</strong> Find n using the sum of binomial coefficients.</p><p>The sum of binomial coefficients: C(n,0) + C(n,1) + ... + C(n,n) = 2^n</p><p>Given: 2^n = 1024 = 2^10</p><p>Therefore: <strong>n = 10</strong></p><p><strong>Step 2:</strong> Find which term has the greatest binomial coefficient.</p><p>For even n, the greatest binomial coefficient is C(n, n/2) = C(10, 5)</p><p><strong>Step 3:</strong> Determine the term number.</p><p>The binomial coefficient C(10, 5) occurs in the (r+1)th term where r = 5</p><p>Therefore, the greatest binomial coefficient occurs in the <strong>(5+1) = 6th term</strong></p><p>∴ Answer: 6</p>
Correct Answer: 6