Question:
If $^{n}C_{12} = ^{n}C_{8}$, then $^{n}C_{17} =$
If $^{n}C_{12} = ^{n}C_{8}$, then $^{n}C_{17} =$
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Always simplify combinations using $^{n}C_r = ^{n}C_{n-r}$ before computing to minimize arithmetic operations ($^{20}C_{17} \to ^{20}C_3$).
Updated On: May 31, 2026
- 1140
- 1150
- 1160
- 1170
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The combination formula satisfies the property that if $^{n}C_{x} = ^{n}C_{y}$, then either $x = y$ or $x + y = n$.
Step 2: Meaning
Since $12 \ne 8$, we must have $n = 12 + 8 = 20$.
Step 3: Analysis
Substitute $n = 20$ into $^{n}C_{17}$: \[ ^{20}C_{17} = ^{20}C_{20-17} = ^{20}C_{3} \] Using the combination formula: \[ ^{20}C_{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 20 \times 19 \times 3 = 1140 \]
Step 4: Conclusion
Therefore, the value of $^{n}C_{17}$ is $1140$. Final Answer: (A)
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