Find $t$ and $n$ knowing that $C_{n+1}^{m+1}: C_{n+1}^{m}: C_{n+1}^{m-1}=5: 5: 3$.

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From the condition, it is clear that $C_{n+1}^{m+1}=C_{n+1}^{m}$, therefore, $m+1=n+1-m$, which means $n=2 m$.

In addition, $\frac{C_{n+1}^{m}}{C_{n+1}^{m-1}}=\frac{n-m+2}{m}=\frac{m+2}{m}=\frac{5}{3}$, so $m=3$.

## Answer

$m=3, n=6$.