Evaluate $^{n^2 + 1}$C$_{n+5}$ if n = 3

1. Analyze the question

We are asked to evaluate the combination formula $^{n^2 + 1}C_{n+5}$ given that $n = 3$.

Recall the general formula for combinations:

2. Explain the concept

To solve this, we first substitute the value of $n$ into the expression to determine the actual values for the upper index ($n^2 + 1$) and the lower index ($n + 5$). Once we have the specific values, we apply the combination formula.

3. Show the solution

Step 1: Substitute $n = 3$ into the expression.

• Upper index: $n^2 + 1 = (3)^2 + 1 = 9 + 1 = 10$
• Lower index: $n + 5 = 3 + 5 = 8$

So, the expression becomes: $^{10}C_8$

Step 2: Apply the combination formula.

Step 3: Simplify the factorials. Expand $10!$ until it reaches $8!$ to cancel it out:

4. Identify the answer

Comparing our result to the given options: A. 55 B. 45 C. 35 D. 25

The correct option is B.

5. Tip

The Symmetry Property: Remember that $^nC_r = ^nC_{n-r}$. In this problem, instead of calculating $^{10}C_8$, you could have calculated $^{10}C_{10-8}$, which is $^{10}C_2$.

This trick saves time and reduces the chance of making errors with large factorials!