If $X\_1$, $X\_2$, $\dots$, $X\_k$ are random variables with Poisson distribution with the following parameters: $\lambda\_1$, $\lambda\_2$, $\dots$, $\lambda\_k$.

Show that $(X\_1 \mid X\_1 + X\_2 + \dots + X\_k)$ (conditional probability) has binomial distribution.

If I suppose that $(X\_1 \mid X\_1 + X\_2 + \dots + X\_k)$ has binomial distribution and given that the sum $Y = X\_1 + X\_2 + \dots + X\_k $ has Poisson distribuition (with $\sum\_i^k \lambda\_i$ parameter), I can build the result and the distribution is a Binomial with parameters $(k, \frac{\lambda\_1}{\lambda\_1 + \lambda\_2 + \dots + \lambda\_k})$, but looks like I done from the end to beginning. There is a direct method?

I am asking for an explanation, and the similar post did not help me.