Question

If \( n^+ C_{n+1} - n^3 C_n = 15(n + 2) \), then \( n = \)

• A. 15
• B. 23
• C. 21
• D. 27

Hint:

For problems involving binomial coefficients, simplify the terms using factorials and known binomial identities to solve for the unknown variable.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We use the Pascal identity \( \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \) to simplify the difference.
Step 2: Detailed Explanation:
Given: \( {^{n+4}}C_{n+1} - {^{n+3}}C_n = 15(n + 2) \).
Note that \( {^{n+3}}C_n = {^{n+3}}C_{(n+3)-n} = {^{n+3}}C_3 \).
Similarly, \( {^{n+4}}C_{n+1} = {^{n+4}}C_{(n+4)-(n+1)} = {^{n+4}}C_3 \).
Equation becomes: \( {^{n+4}}C_3 - {^{n+3}}C_3 = 15(n + 2) \).
Using Pascal property \( \binom{n+3}{3} + \binom{n+3}{2} = \binom{n+4}{3} \):
\( {^{n+4}}C_3 - {^{n+3}}C_3 = {^{n+3}}C_2 \).
So, \( {^{n+3}}C_2 = 15(n + 2) \).
\( \frac{(n + 3)(n + 2)}{2 \times 1} = 15(n + 2) \).
Cancelling \( (n + 2) \) as \( n \neq -2 \):
\( \frac{n + 3}{2} = 15 \Rightarrow n + 3 = 30 \Rightarrow n = 27 \).
Step 3: Final Answer:
The value of \( n \) is 27.