Let:

$$p_1 ,p_2, \ldots, p_k$$

...be positive integers and suppose that:

$$p_1+p_2+\cdots +p_k-k+1$$

...pigeons have $k$ pigeonholes to fly into.

Which of the following statements must be true?

I) For:

$$i=1,2,\ldots,k$$

...the $i$-th pigeonhole contains at least $p_i$ pigeons.

II) For:

$$o=1,2,\ldots,k$$

...the $i$-th pigeonhole contains at least $p_i-1$ pigeons.

III) There is an $i$ such that the $i$-th pigeonhole contains at least $p_i$ pigeons.

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