Why $H_5(\mathbb{R}P^5) = \mathbb{Z}$?

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Why $H_5(\mathbb{R}P^5) = \mathbb{Z}$?

Can anyone describe it for me using the formula $H_{n} = \frac {\ker{\partial_{n}}}{Im{\partial_{n+1}}}.$ I know that this is correct because it is orientable but I need a proof by the above mentioned formula.