Question

For the positive integer $ n $, $ C_1^{n} + C_2^{n} + C_3^{n} + ... + C_n^{n} \text{ is equal to} $

• A. \( 2^n \)
• B. \( 2^n - 1 \)
• C. \( n^2 \)
• D. \( n^2 - 1 \)

Hint:

When dealing with sums of binomial coefficients, remember that the sum of all binomial coefficients \( \binom{n}{k} \) from \( k = 0 \) to \( n \) is equal to \( 2^n \). Subtracting the first term gives the desired result.

Answer 1

The Correct Option is B

Step 1: Understand the notation.
The given expression is: \[ C_1^{n} + C_2^{n} + C_3^{n} + \dots + C_n^{n} \] Where \( C_k^{n} \) refers to the binomial coefficient \( \binom{n}{k} \).
This is the sum of binomial coefficients raised to the power of \( n \).
Step 2: Apply the binomial expansion.
The sum \( C_1^n + C_2^n + C_3^n + \dots + C_n^n \) is a form of a binomial expansion that arises from the expansion of the binomial series: \[ (1 + 1)^n = 2^n \] This is derived from the binomial theorem, which tells us that the sum of the coefficients (i.e., \( \binom{n}{k} \) for each \( k \)) in a binomial expansion sums to \( 2^n \).
Thus, the sum of the binomial coefficients \( C_k^n \) from \( k = 1 \) to \( n \) is related to the total binomial sum.
Step 3: Subtract the first term.
We subtract 1 (the first term of the binomial expansion for \( k = 0 \)) since we start from \( C_1^n \): \[ C_1^n + C_2^n + C_3^n + \dots + C_n^n = 2^n - 1 \]
Step 4: Conclusion.
Therefore, the sum \( C_1^n + C_2^n + C_3^n + \dots + C_n^n \) is equal to \( 2^n - 1 \), which corresponds to option (b).