Cn^3=4/15*C(n+2)^4

To simplify the given equation \( Cn^3 = \frac{4}{15}C(n+2)^4 \), we first expand the right side by multiplying the terms inside the parentheses:

\[ Cn^3 = \frac{4}{15}C(n^4 + 8n^3 + 16n^2 + 32n) \]

Next, distribute \( \frac{4}{15} \) to each term inside the parentheses:

\[ Cn^3 = \frac{4}{15}Cn^4 + \frac{32}{15}Cn^3 + \frac{64}{15}Cn^2 + \frac{128}{15}Cn \]

Now, we can compare the terms on both sides of the equation:

• Coefficients of \( n^3 \): 1 = \( \frac{32}{15}C \)
• Coefficients of \( n^2 \): 0 = \( \frac{64}{15}C \)
• Coefficients of \( n \): 0 = \( \frac{128}{15}C \)
• Constant terms: 0 = 0

From the first equation:

\[ \frac{32}{15}C = 1 \]

Solving for \( C \):

\[ C = \frac{15}{32} \]

Therefore, the value of \( C \) is \( \frac{15}{32} \).