If the sum of binomial coefficients in the expansion of $(x+y){}^n$ is 256, then the greatest binomial coefficient occurs in which one of the following terms?

Concept:

Sum of Binomial Coefficients and Greatest Binomial Coefficient:

• The sum of binomial coefficients in the expansion of \( (x + y)^n \) is calculated by substituting x = 1 and y = 1. The result is \(2^n\).
• To find the greatest binomial coefficient, we analyze the coefficients \(C(n, r)\) where r is the term index in the expansion. The greatest coefficient occurs near the middle term(s).
• Key Formulae:
  • Sum of binomial coefficients: \( \text{Sum} = 2^n \)
  • Binomial coefficient: \( C(n, r) = \frac{n!}{r!(n-r)!} \)
  • Greatest binomial coefficient: For even n, it occurs at r = n/2. For odd n, it occurs at r = (n-1)/2 and r = (n+1)/2.

Calculation:

Given,

Sum of binomial coefficients = \(2^n = 256\)

We calculate n:

\( 2^n = 256 \)

⇒ \(2^8 = 256\)

Greatest Binomial Coefficient:

For \( n = 8 \) (even), the greatest binomial coefficient occurs at \( r = n/2 = 8/2 = 4 \).

⇒ The term index is r = 4, which corresponds to the 5th term (since indexing starts from 0).

∴ The greatest binomial coefficient occurs in the 5th term.