Question

The value of the sum \( 15 C_6 + 14 C_6 + 13 C_6 + 12 C_6 + 11 C_6 + 10 C_6 \) is equal to:

• A. \( 15 C_7 - 10 C_6 \)
• B. \( 15 C_7 - 10 C_7 \)
• C. \( 16 C_7 - 10 C_7 \)
• D. \( 16 C_7 - 10 C_6 \)
• E. \( 16 C_7 - 11 C_6 \)

Hint:

In problems involving binomial coefficients, remember to simplify the terms carefully and look for any common patterns or factorization that can help in matching the correct option.

Answer 1

The Correct Option is C

We are given the sum: \[ S = 15 C_6 + 14 C_6 + 13 C_6 + 12 C_6 + 11 C_6 + 10 C_6 \] This can be simplified by factoring out \( C_6 \): \[ S = C_6 \times (15 + 14 + 13 + 12 + 11 + 10) \] Now, calculating the sum of the numbers inside the parentheses: \[ 15 + 14 + 13 + 12 + 11 + 10 = 75 \] So, the expression becomes: \[ S = 75 C_6 \] Next, we observe that the binomial coefficients in the options suggest a shift in the terms. We know that: \[ C_7 = C_6 + C_6 \] Thus, the correct simplification for the given sum is \( 16 C_7 - 10 C_7 \), which matches option (C).
Thus, the correct answer is option (C), \( 16 C_7 - 10 C_7 \).