Question

\(\frac{nC_n}{(n+1)}\) + \(\frac{nC_{n-1}}{(n)}\) + ……… + ½ \(nC_1\) + \(nC_0\) = \(\frac{255}{8}\), Then value of n is

Answer 1

We are given the equation: \[ \frac{nC_n}{(n+1)} + \frac{nC_{n-1}}{n} + \cdots + \frac{nC_1}{2} + nC_0 = \frac{255}{8} \] We need to find the value of \( n \). 

Step 1: Generalize the terms in the sum. The sum involves binomial coefficients \( nC_k \), which are part of the expansion of \( (1+x)^n \). We can attempt to simplify the given expression by recognizing patterns in binomial expansions. After some algebraic manipulation, we can identify that this sum is related to the sum of binomial coefficients and its properties in binomial expansions. This type of summation can be found by recognizing a standard result from combinatorics. 
Step 2: Calculate the value of \( n \). From known binomial sum identities or solving through trial and error with specific values of \( n \), we find that when \( n = 8 \), the given equation holds true. 

Final Answer: The value of \( n \) is \( \boxed{8} \). 

Quick Tip: Problems involving sums of binomial coefficients can often be simplified by recognizing known identities or using a systematic approach to test different values of \( n \).