Is it permissible to slaughter a sheep during Hajj with the intention of it being an expiation for not spending the night in Mina and an 'Aqiqah for one of the children simultaneously?

The predominant view is that whoever fails to spend the nights in Mina must offer a sacrifice (dam). It is not permissible for him to combine it with the 'Aqiqah (definition) (sacrifice for a newborn) with a single intention, because each of them is an act of worship intended for its own sake and has a different cause. The intermingling between them invalidates the purpose of each. Since combining the 'Udhiyah (definition) (sacrifice for Eid al-Adha) and 'Aqiqah is prohibited, prohibiting the combination of 'Aqiqah and the dam required for failing to spend the nights in Mina is even more appropriate.