If nC10 is equal to nC12 for some positive integer n, then what is the value of n that satisfies this binomial coefficient equality?

Introduction / Context:
This question tests knowledge of binomial coefficients and symmetry properties of nCr. When two binomial coefficients with the same n but different r values are equal, there is a specific relationship between the r values and n. Understanding this relationship is important in many combinatorial proofs and problems.

Given Data / Assumptions:

• We are given that nC10 = nC12 for some integer n.
• We assume n is a positive integer and at least as large as the larger r, so n is at least 12.
• We want to determine the value of n that makes these two coefficients equal.

Concept / Approach:
A key property of binomial coefficients is the symmetry nCr = nC(n - r). If two coefficients with same n are equal and r values are different, then typically one of them equals n minus the other. In other words, if nCr1 = nCr2 and r1 is not equal to r2, we usually have r1 + r2 = n. We apply this concept directly here with r1 = 10 and r2 = 12.

Step-by-Step Solution:
Step 1: Use the property: if nCr1 = nCr2 and r1 is not equal to r2, then r1 + r2 = n.Step 2: Here r1 = 10 and r2 = 12.Step 3: Add the r values: 10 + 12 = 22.Step 4: Therefore, n must be 22 to satisfy nC10 = nC12.Step 5: Optionally verify using symmetry: for n = 22, nC10 = 22C10 and nC12 = 22C(22 - 10) = 22C12. Because 10 and 12 sum to 22, the symmetry property guarantees equality.

Verification / Alternative check:
As an algebraic check, you could write nC10 and nC12 using factorials and equate them. After cancelling common factorial factors, the equation simplifies and leads to n = 22. This process is longer but matches the quick symmetry based method and confirms that there is a unique positive integer solution for n in this context.

Why Other Options Are Wrong:

• 10 and 12: These are values of r, not n. Choosing them confuses the role of n and r in the binomial coefficient.
• 24: This would make r1 + r2 less than n, and 24C10 would equal 24C14, not 24C12, so the equality would not hold.

Common Pitfalls:
A common mistake is to try to expand the factorials fully, which is time consuming and error prone. Another is to think that the equality holds for any n greater than or equal to 12, which is not true. Some students also incorrectly apply properties like nCr = nC(r + 1), which do not hold. Remembering the correct symmetry relation nCr = nC(n - r) and using r1 + r2 = n is the clean way to handle such problems.

Final Answer:
The value of n for which nC10 = nC12 is 22.