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:
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:
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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.
Updated On: Apr 7, 2026
- \( 15 C_7 - 10 C_6 \)
- \( 15 C_7 - 10 C_7 \)
- \( 16 C_7 - 10 C_7 \)
- \( 16 C_7 - 10 C_6 \)
- \( 16 C_7 - 11 C_6 \)
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The Correct Option is C
Solution and Explanation
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), \( 16 C_7 - 10 C_7 \).
Thus, the correct answer is option (C), \( 16 C_7 - 10 C_7 \).
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